Turbulent Dynamics in Rotating Helium Superfluids€¦ · Turbulent Dynamics in Rotating Helium Superfluids 47 superﬂuid order parameter ﬁeld, a transition from linear line-like

CHAPTER2Turbulent Dynamics in RotatingHelium SuperfluidsV. B. Eltsov*, R. de Graaf*, R. Hänninen*,M. Krusius*, R. E. Solntsev*, V. S. L'vov†,A. I. Golov‡ and P. M. Walmsley‡

Contents 1. Introduction 462. Dynamic Instability–Precursor to Turbulence 50

2.1 Introduction 502.2 Seed Vortex Evolution in Rotating Column 542.3 Onset Temperature of Turbulence 592.4 Single-Vortex Instability in Applied Flow 642.5 Numerical Calculation of Dynamic Vortex

Generation 682.6 Summary: Onset of Turbulence 72

3. Propagating Vortex Front in Rotating Flow 733.1 Introduction 733.2 Measurement of Vortex Front Propagation 753.3 Velocity of vortex front 793.4 Numerical Calculation of Turbulence in Vortex

Front Propagation 853.5 Analytical model of turbulent front 963.6 Summary: Turbulent Vortex Front Propagation 111

4. Decay of Homogeneous Turbulence inSuperfluid 4He 1124.1 Introduction and Experimental Details 1124.2 Experimental Results 1244.3 Discussion: Dissipation in Different Types of

Turbulence 1324.4 Summary: Decay of Turbulence on Quasiclassical and

Ultraquantum Scales 1395. Summary 141Acknowledgements 141References 142

* Low Temperature Laboratory, Helsinki University of Technology, P.O.Box 5100, FI-02015-TKK, Finland†Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel‡School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK

Progress in Low Temperature Physics: Quantum Turbulence,Volume 16 © 2009 Elsevier B.V.ISSN 0079-6417, DOI 10.1016/S0079-6417(08)00002-4 All rights reserved.

45

46 V. B. Eltsov et al.

Abstract New techniques, both for generating and detecting turbulence inthe helium superfluids 3He-B and 4He, have recently given insight inhow turbulence is started, what the dissipation mechanisms are, andhow turbulence decays when it appears as a transient state or whenexternally applied turbulent pumping is switched off. Importantsimplifications are obtained by using 3He-B as working fluid, wherethe highly viscous normal component is practically always in astate of laminar flow, or by cooling 4He to low temperatures wherethe normal fraction becomes vanishingly small. We describe recentstudies from the low temperature regime, where mutual frictionbecomes small or practically vanishes. This allows us to elucidatethe mechanisms at work in quantum turbulence on approaching thezero temperature limit.

1. INTRODUCTION

The transition to turbulence is the most well known example of all hydro-dynamic transitions. It has been marveled for centuries since dramaticdemonstrations can be seen everywhere where a sudden change in theflow occurs, owing to a constriction in the flow geometry, for instance.For 50 years, it has been known that turbulence also exists in superflu-ids (Vinen and Donnelly, 2007), although by its very nature a superfluidshould be a dissipation-free system. In many situations, it is found on themacroscopic level that superfluid vortex dynamics mimics the responses ofviscous hydrodynamics. This is one of the reasons why it has been thoughtthat superfluid turbulence might provide a shortcut to better understand-ing of turbulence in general. From the developments over the past 50 yearswe see that this has not become the case, superfluid turbulence is a com-plex phenomenon where experiments have often been clouded by otherissues, especially by vortex formation and vortex pinning. Nevertheless,the topic is fascinating in its own right: when the flow velocity is increased,the inherently dissipation-free superfluid is observed to become dissipa-tive and eventually turbulent. This is particularly intriguing in the zerotemperature limit where the density of thermal excitations approacheszero and vortex motion becomes undamped down to very short wavelengths (of the order of the vortex core diameter).

There are two isotropic helium superfluids in which turbulence hasbeen studied, namely the B phase of superfluid 3He (3He-B) and super-fluid 4He (4He II). In the anisotropic A phase of superfluid 3He (3He-A),dissipation is so large that conventional superfluid turbulence is notexpected at the now accessible temperatures above 0.1 Tc (Finne et al.,2003). Instead rapid dynamics and large flow velocities promote in 3He-Aa transition in the topology and structure of the axially anisotropic

Turbulent Dynamics in Rotating Helium Superfluids 47

superfluid order parameter field, a transition from linear line-like vor-tices to planar sheet-like vortices (Eltsov et al., 2002). Turbulence has alsobeen studied in laser-cooled Bose-Einstein condensed cold atom cloudsalthough so far only theoretically (Kobayashi and Tsubota, 2008; Parkerand Adams, 2005), but it is expected that experiments will soon follow.Here, we are reviewing recent work on turbulence in rotating flow inboth 3He-B and 4He II, emphasising similarities in their macroscopicdynamics.

A number of developments have shed new light on superfluid turbu-lence. Much of this progress has been techniques driven in the sense thatnovel methods have been required, to make further advances in a fieldas complex as turbulence, where the available techniques both for gener-ating and detecting the phenomenon are not ideal. Three developmentswill be discussed in this review, namely (i) the use of superfluid 3He forstudies in turbulence, which has made it possible to examine the influenceof a different set of superfluid properties in addition to those of superfluid4He, (ii) the study of superfluid 4He in the zero temperature limit wherethe often present turbulence of the normal component does not complicatethe analysis and (iii) the use of better numerical calculations for illustrationand analysis.

From the physics point of view, three major advances can be listed toemerge: in superfluid 3He, one can study the transition to turbulence as afunction of the dissipation in vortex motion (Eltsov et al., 2006a), known asmutual friction. The dissipation arises from the interaction of thermal exci-tations with the superfluid vortex when the vortex moves with respect tothe normal component. In classical viscous flow, such a transition to turbu-lence would conceptually correspond to one as a function of viscosity. Thisis a new aspect, for which we have to thank the 3He-B Fermi superfluidwhere the easily accessible range of variation in mutual friction dissipa-tion is much wider than in the more conventional 4He II Bose superfluid.We are going to make use of this feature in Section 2 where we examine theonset of superfluid turbulence as a function of mutual friction dissipation(Finne et al., 2006a).

Second, in Section 3, we characterise the total turbulent dissipation insuperfluid 3He as a function of temperature, extracted from measurementsof the propagation velocity of a turbulent vortex front (Eltsov et al., 2007).A particular simplification in this context is the high value of viscosityof the 3He normal component, which means that in practice the normalfraction always remains in a state of laminar flow.

Finally, our third main topic in Section 4 are the results from recention transmission measurements in superfluid 4He (Walmsley et al., 2007a;Walmsley and Golov, 2008a), where the decay of turbulence is recordedfrom 1.6 K to 0.08 K. Here turbulent dissipation can be examined in the truezero temperature limit with no normal component. As a result we now


know that turbulence and dissipation continue to exist at the very lowesttemperatures. Although the dissipation mechanisms of 4He or 3He-B inthe T → 0 limit are not yet firmly established (Kozik and Svistunov, 2005b;Vinen, 2000, 2001), it is anticipated that these questions will be resolved inthe near future (Kozik and Svistunov, 2008b; Vinen, 2006).

Phrased differently, our three studies address the questions (i) howturbulence starts from a seed vortex which is placed in applied vortex-free flow in the turbulent temperature regime (Section 2), (ii) how vorticesexpand into a region of vortex-free flow (Section 3) and (iii) how the vor-ticity decays when the external pumping is switched off (Section 4). Thecommon feature of these three studies is the use of uniformly rotatingflow for creating turbulence and for calibrating the detection of vortic-ity. Turbulence can be created in a superfluid in many different ways,but a steady state of constant rotation does generally not support tur-bulence. Nevertheless, at present rotation is the most practical meansof applying flow in a controlled manner in the millikelvin tempera-ture range. In this review, we describe a few ways to study turbulencein a rotating refrigerator. Superfluid hydrodynamics supports differentkinds of flow even in the zero temperature limit so that turbulent lossescan vary greatly both in form and in magnitude, but generally speak-ing, the relative importance of turbulent losses tends to increase withdecreasing temperature. Two opposite extremes will be examined: highlypolarised flow of superfluid 3He-B when a vortex front propagates alonga rotating cylinder of circular cross-section (Section 3), and the decay of anearly homogeneous isotropic vortex tangle in superfluid 4He (Section 4),created by suddenly stopping the rotation of a container with squarecross-section.

Turbulent flow in superfluid 3He-B and 4He is generally described bythe same two-fluid hydrodynamics of an inviscid superfluid componentwith singly-quantised vortex lines and a viscous normal component. Thetwo components interact via mutual friction. There are generic proper-ties of turbulence that are expected to be common for both superfluids.However, there are also interesting differences which extend the rangeof the different dynamic phenomena which can be studied in the Hesuperfluids:

• In typical experiments with 3He-B, unlike with 4He, the mutual frictionparameter α can be both greater and smaller than unity (Figure 1) – thisallows the study of the critical limit for the onset of turbulence at α ∼ 1(Section 2).

• The viscosity of the normal component in 3He-B is four orders of magni-tude higher than in 4He; hence the normal component in 3He-B is rarelyturbulent, which amounts to a major simplification at finite temperatures(but not in the T = 0 limit with a vanishing normal component).


0.2 0.3 0.4 0.5 0.6 0.7 0.8

T/Tc

0.1

1

104

103

102

101

(12

a′)/

a3He-B (29 bar)Bevan et al. (1997)

105

104

103

102

101(12

a′)/

a

T (K)

0.8 1.2 1.6 20.6 1 1.4 1.8 2.2

4He (svp)Donnelly & Barenghi (1998)

1

0.1

FIGURE 1 Mutual friction parameter ζ = (1 − α′)/α as a function of temperature. Insuperfluid dynamics, this parameter, composed of the dissipative mutual friction α(T )and the reactive mutual friction α′(T ), corresponds to the Reynolds number Re ofviscous hydrodynamics. Typically, when Re > 1, turbulence becomes possible in thebulk volume between interacting evolving vortices. This transition to turbulence as afunction of temperature can readily be observed in 3He-B (at 0.59 Tc), while in 4He II,it would be within ∼0.01 K from the lambda temperature and has not beenidentified yet.

• While the critical velocity vc for vortex nucleation is much smaller in3He-B, pinning on wall roughness is also weaker; this makes it possibleto create vortex-free samples which are instrumental in the transitionalprocesses studied in Sections 2 and 3; on the other hand, the ever-presentremanent vortices in superfluid 4He are expected to ease the productionof new vortices, which becomes important in such experiments as spin-up from rest.

• The vortex core diameter in both liquids is small (which allows us touse the model of one-dimensional line filaments), but in 3He-B, it is upto three orders of magnitude larger than in 4He; hence the dissipationmechanisms in the T → 0 regime, which ultimately rely on the emissionof excitations, are expected to work in 3He-B at larger length scales andto lead to more significant energy loss in vortex reconnections.

A comparison of the turbulent dynamics in these two superfluidsallows one to identify generic properties that are common for both super-fluids, and also to pinpoint specific reasons when there are differences.The main quantity controlling dissipation is the mutual friction dissipa-tion α(T), which dominates the temperature dependence of the dynamicmutual friction parameter ζ = (1 − α′)/α, shown in Figure 1. Experimentalvalues are plotted with filled symbols for 3He-B (Bevan et al., 1997) andwith a solid line for 4He (Donnelly and Barenghi, 1998). At low tempera-tures, the following extrapolations are used (shown as dashed lines): for3He-B at a pressure P = 29 bar, we use α = 37.21 exp(−1.968Tc/T), where


the value for the superfluid gap � = 1.968Tc is a linear interpolation asa function of density ρ between the weak coupling value at zero pres-sure and that measured by Todoshchenko et al. (2002) at melting pressure.For 4He at saturated vapour pressure (svp), we follow Kozik and Svis-tunov (2008b) and use α = 25.3 exp(−8.5/T)T−1/2 + 5.78 · 10−5T5, whereT is in K.

2. DYNAMIC INSTABILITY–PRECURSOR TO TURBULENCE

2.1 Introduction

In practice, superfluid flow remains dissipation-free only as long as thereare no quantised vortices (or the existing vortices do not move, whichis more difficult to arrange). The classic question in superfluid hydrody-namics is therefore: how is the quantised vortex formed (Feynman, 1955;Vinen, 1963)? In flow measurements with bulk liquid, the understandingabout the origin of the first vortex has been improving in recent times.Whether it is created in an intrinsic nucleation process (Ruutu et al., 1997)or from remanent vortices (Solntsev et al., 2007), which were created ear-lier in the presence of flow or while cooling the sample to the superfluidstate (Hashimoto et al., 2007), these questions we are not going to addresshere. Instead we assume that the first vortex is already there, for instanceas a remanent vortex. We then ask the question: how is turbulence startedwhen the flow velocity is suddenly increased by external means? Afterall a turbulent vortex tangle is created through the interaction of manyvortices: so how can turbulence start from a single seed vortex?

In rotating 3He-B, one can create reliably a metastable state of vortex-free flow. It is then possible to inject a single vortex ring into the flow withneutron irradiation. When a slow thermal neutron undergoes a capturereaction in liquid 3He with a 3He nucleus, a vortex ring may escape fromthe overheated reaction bubble into the flow if the flow velocity is above acritical value (Eltsov et al., 2005). Making use of this phenomenon, one caninject a single vortex ring in vortex-free flow at different temperatures. Atlow temperatures, it is observed that a turbulent vortex tangle is sponta-neously formed from the injected ring (Finne et al., 2004a), while at hightemperatures, only a single vortex line results (Ruutu et al., 1998a). Whatis the explanation?

This demonstration in 3He-B shows that in addition to the appliedflow velocity also mutual friction matters importantly in the formation ofnew vortices, in their expansion, and in the onset of turbulence. In 3He-B, mutual friction dissipation α(T) is strongly temperature-dependent(Figure 1), and it so happens that α(T) drops to sufficiently low value forthe onset of turbulence in the middle of the accessible temperature range.The principle of seed vortex injection experiments is summarised in


Cooling throughTc at Ω 5 const

Equilibrium vortex state 2 Minimum energy state

Vortex-free counterflow 2 Maximum energy state

Localizedturbulent burst

T/Tc

Tc0.2 0.4 0.6 0.8

Cooling throughTc at Ω 5 0 fol-lowed by rota-tion at Ω < �c/R

Injection of well-separated seed vortices

z5 (12a9)/a0.1110K

inet

ic e

nerg

y

Slowly movingspatially extended

turbulence

Transition not possibleat Ω 5 const

4He-II

100

FIGURE 2 Principle of measurements on seed vortex injection at constant rotation �

and temperature T . Well-separated isolated seed vortices are introduced in rotatingvortex-free counterflow. The initial high-energy state may then relax to theequilibrium vortex state via vortex generation processes which become possible attemperatures below the hydrodynamic transition at 0.59 Tc. The Kelvin-waveinstability of a single evolving seed vortex is the first step in this process. It is thenfollowed by a turbulent burst which is started if the density of newly created vorticesgrows sufficiently. The combined process depends on the dynamic mutual frictionparameter ζ = (1 − α′)/α which is shown on the top. On the very top, the range ofvariation for this parameter in 4He-II is indicated, that is, the temperature regime ofconventional 4He-II measurements.

Figure 2. In these measurements, the number and configuration of theinjected vortex loops can be varied. It turns out that the highest transi-tion temperature is observed when turbulence starts in bulk volume frommany small vortex loops in close proximity of each other. This transitionhas been found to be at Tbulk

on ∼ 0.6 Tc (Finne et al., 2004b) and to be inde-pendent of flow velocity over a range of velocities (3 – 6 mm/s). In viscoushydrodynamics, the Reynolds number is defined as Re = UD/νcl, whereU is the characteristic flow velocity, D the relevant length scale of theflow geometry, and νcl = η/ρ the kinematic viscosity. In an isotropic super-fluid, the equivalent of the Reynolds number proves to be ζ = (1 − α′)/α. Itdefines the boundary between laminar and turbulent flow as a function ofdissipation and is independent of flow velocity or geometry (Finne et al.,2006b).

However, if we inject instead of several closely packed vortex loopsonly one single seed loop in vortex-free flow (or several loops but so farapart that they do not immediately interact), then the transition to tur-bulence is found to be at a lower temperature and to depend on the flowvelocity. Thus the onset of turbulence must also have a velocity dependentboundary.


To explain all these observations, one has to assume that an indepen-dent precursor mechanism exists, which creates more vortices from a seedvortex evolving in externally applied flow. The characterisation of thisinstability is the topic of this section. It turns out that this can be done in3He-B in the temperature regime close to the onset Tbulk

on of turbulence inthe bulk volume. Here the precursor often progresses sufficiently slowly sothat it can be captured with present measuring techniques, while at lowertemperatures turbulence starts too rapidly. This latter case is exactly whathappens in 4He II: mutual friction dissipation is always so low in the usualexperimental temperature range that the instability has not been explicitlyidentified.

The central question is the reduced stability of the evolving seed vortexloop when mutual friction dissipation is decreasing on cooling to lowertemperature. At sufficiently low α(T), an evolving vortex becomes unsta-ble with respect to loop formation, so that one or more new vortex loopsare split off, before the seed vortex has managed to reach its stable stateas a rectilinear line parallel to the rotation axis. The evolution during thisentire process, from injection to the final state, is depicted in Figure 3in a rotating cylindrical sample. The final state is the equilibrium vortexstate, with an array of rectilinear vortex lines, where their areal densitynv in the transverse plane is given by the rotation velocity �: nv = 2�/κ.Here, κ = 2π�/2m3 = 0.066 mm2/s is the superfluid circulation quantumof the condensate with Cooper-pairs of mass 2m3. In this equilibrium stateat constant rotation, the superfluid component is locked to solid-body

time

Vortex frontand

twisted state

Relaxingtwist

Single-vortexinstability

Ω

Equilibriumvortex state

ΩΩΩ

Turbulentburst

Ω

�n 5 Ω × r

→

Seed vortexin

rotating flow

Ω

�s5 0

Seed

→

≈ < �s >

�n 5 Ω × r

Instability

FIGURE 3 Vortex instability and turbulence in a rotating column of 3He-B in theturbulent temperature regime, T < T bulk

on . A seed vortex loop is injected in appliedvortex-free flow and the subsequent evolution is depicted. Different transient statesare traversed until the stable rotating equilibrium vortex state is reached.


rotation with the normal component when averaged over lengths exceed-ing the inter-vortex distance � ∼ 1/

√nv. In the ideal case, all vortices are

here rectilinear, while in the transient states in Figure 3, vortices can existin many different configurations.

In Figure 4, a rough classification is provided of the stability of vor-tices as a function of temperature (or more exactly mutual friction) indifferent configurations and rotating situations. The lowest temperaturesbelow 0.3 Tc are in the focus of current research and have by now beenprobed with a few different types of measurements. The most extensivework has been performed by the Lancaster group. They create with vari-ous vibrating oscillators in a quiescent 3He-B bath a vortex tangle and thenmonitor the decay of the tangle with a vibrating wire resonator (Bradleyet al., 2006). The total turbulent dissipation in a vortex front propagatingalong a rotating column (see Figure 3) has recently been measured (Eltsovet al., 2007) and will be discussed in Section 3. Also the response of the

T/Tc

Tc0.2 0.4 0.6 0.8

1.80.650.14 a(T)

4He-II

Ideal regularvortex formation

at �c

Vortex number con-served at Ω 5 const

Bulk turbulence exists

Rectilinear vortices are stable“Rectilinear” vorticesdestabilized by Ω 5 0?

Evolving vortexalways unstable

Onset regime ofdynamic vortex

generation

FIGURE 4 Summary of vortex stability in rotating counterflow of 3He-B, as afunction of temperature. Top row: the hydrodynamic transition at Tbulk

on ≈ 0.59 Tc (atP = 29 bar pressure) separates regular and turbulent vortex dynamics. Above thetransition, vortices are stable in all situations which have been studied, while belowthe transition, turbulence becomes possible. Middle row: in rotation at constant �,rectilinear vortices are stable. In time-dependent rotation (|�| �= 0), the “rectilinear”vortex turns out to be an idealisation, presumably because of the experimentallyinevitable slight misalignment between the rotation and the sample cylinder axes andbecause of surface interactions. In practice, the “rectilinear” vortices are found to bestable above ∼0.3 Tc in time-dependent rotation, while at lower temperatures, theytend to transform to increasingly turbulent configurations with increasing |�|. Bottomrow: dynamically evolving vortices are stable above the transition, but at lowertemperatures, an evolving vortex may become unstable, generate a new vortex andeventually bulk turbulence. The conditions at seed vortex injection determine theonset temperature Ton below which turbulence follows. The onset temperatures havebeen found to concentrate in the regime 0.35 Tc < Ton < 0.59 Tc. The very lowtemperatures below 0.3 Tc display consistently turbulent response.


superfluid component has been studied to rapid step-like changes in rota-tion when � is changed from one constant value to another. This type ofmeasurement is commonly known as spin-up or spin-down of the super-fluid fraction and will be extensively described for spin-down in the caseof superfluid 4He later in this review.

2.2 Seed Vortex Evolution in Rotating Column

The motion of a seed vortex follows a distinctive pattern while it expandsin a rotating cylinder. Numerically calculated illustrations are shown inFigures 5 and 6 which depict the evolution of the seeds to stable rectilinearlines. In Figure 5, an example with remanent vortices is examined, while inFigure 6, the initial configuration is an equilibrium vortex state in the usualsituation that the rotation and cylinder axes are inclined by some small

t 5 0N 5 22 vorticesΩ 5 0.1 rad/s

t 5 600 sN 512Ωi5 0

t 5 700 sN 512

Ωf5 0.5 rad/s

t 5 800 sN 512

Ωf5 0.5 rad/s

FIGURE 5 Numerical calculation of the evolution of remanent vortices in rotatingflow (de Graaf et al., 2007). t ≤ 0: initial state with 22 vortices at 0.1 rad/s rotation.The vortices have been artificially tilted by 1.4◦, by displacing their end pointsuniformly by 1mm at both end plates of the cylinder, to break cylindrical symmetry.t = 0: rotation is abruptly reduced to zero to allow vortices to annihilate. t = 600 s:after a waiting period �t = 600 s at zero rotation, 12 remanent vortices remain indynamic state. Rotation is then suddenly increased to �f = 0.5 rad/s. t ≥ 600 s: the 12remnants start evolving towards rectilinear lines. This requires that the vortex ends onthe cylindrical wall travel in spiral motion to the respective end plates. Theparameters are radius R = 3mm and length h = 80mm of cylinder, T = 0.4 Tc,P = 29.0 bar, α = 0.18 and α′ = 0.16 (Bevan et al., 1997). In the figure, the radial lengthshave been expanded by two compared to axial distances.


t ≤ 0Ωi5 0.03 rad/s

60 sΩf5 0.2 rad/s

90 sΩf

120 sΩf

FIGURE 6 Numerical calculation of the evolution of two curved peripheral vorticesin an inclined rotating cylinder, when rotation is suddenly increased at t = 0 from theequilibrium vortex state at �i ≈ 0.03 rad/s to �f = 0.2 rad/s (Hänninen et al., 2007a).There are 22 vortices in this sample, of which two in the outermost ring (lyingopposite to each other) have been initially bent to the cylindrical wall, to mimic aninclined cylinder. In the later snapshots at �f, the two short vortices expand towardsthe top and bottom end plates of the cylinder, to reach their final stable state asrectilinear lines. Parameters: R = 3mm, h = 30mm, P = 29.0 bar and T = 0.4 Tc (whichcorresponds to α = 0.18 and α′ = 0.16).

residual angle ∼1◦. These calculations (in a rotating circular cylinder withradius R and length h) describe the situation at intermediate temperatureswhen the vortex instability does not yet occur. The purpose is to focuson the motion of the expanding vortices and the transient configurationswhich thereby evolve. The characteristic property is the spiral trajectoryof a vortex end along the cylindrical wall and the strong polarisation onan average along the rotation axis. The calculations have been performedusing the numerical techniques described in Section 2.5 (de Graaf et al.,2007).

In Figure 5, the remnants are obtained from an equilibrium vortex staterotating at 0.1 rad/s, by reducing rotation to zero in a sudden step-likedeceleration. The vortices are then allowed to annihilate at zero rotationfor a period �t. Some leftover remnants, which have not yet managed toannihilate, still remain after this annihilation time. Ideally smooth walls


are assumed without pinning. By suddenly increasing rotation from zeroto a steady value �f = 0.5 rad/s, the remaining remnants are forced toexpand. The configuration 100 s later shows how the spiral vortex motionhas created a twisted vortex cluster in the centre, with a few vortex endsstill traveling in circular motion around the cluster. This motion thus windsthe evolving vortex around the straighter vortices in the centre. On the farright 200 s after the start of the expansion, the cluster is almost completed.Nevertheless, this state is still evolving since ultimately also the helicaltwist relaxes to rectilinear lines, while the vortex ends slide along the endplates of the container.

In Figure 6, a calculation is presented with 20 rectilinear vortex lines andtwo short vortices which connect at one end to the cylindrical wall. Thisconfiguration mimics the equilibrium vortex state in a real rotating experi-ment where there exists some residual misalignment between the rotationand sample cylinder axes. Depending on the angle of misalignment and theangular velocity of rotation �i, some of the peripheral vortices may thenend on the cylindrical sidewall in the equilibrium vortex state, as shownon the far left. At t = 0, rotation is increased in step-like manner from �i toa higher value �f. Two types of vortex motion are started by the rotationincrease. First, the N = 20 rectilinear vortices are compressed to a centralcluster with an areal density nv = 2�f/κ by the surrounding azimuthallyflowing counterflow. Outside the vortex cluster, the counterflow has thevelocity

v(�f, r, N) = vn − vs = �fr − κN/(2πr). (1)

The normal excitations are in solid-body rotation and thus vn = �fr, whilethe superflow velocity around a cluster of κN circulation quanta is decay-ing as vs = κN/(2πr), where r ≥ Rv and the cluster radius Rv ≈ R

√�i/�f.

Experimentally it is convenient to define the number of vortices N viathe initial equilibrium vortex state: N = Neq(�i), where the externallyadjusted rotation velocity �i ∼ κNeq/(2πR2) uniquely defines Neq in agiven experimental setup. It is customary to denote this specially preparedcalibration value with �v = �i. The maximum counterflow velocity at �fat the cylinder wall is then given by v ≈ (�f − �v)R. This definition hasbeen used in Figure 7 to characterise the number of vortices in the centralcluster.

The second type of vortex motion, which is enforced by the increasedrotation in Figure 6, is the spiral motion of the two short vortices asthey become mobile and start expanding towards the top and bot-tom end plates, respectively. Let us now examine this motion in moredetail.

A vortex moving with respect to the superfluid component is subjectto the influence from the Magnus lift force. This force can be written as


47 47.5 48 48.5

Magnetic field sweep (mT)

0

0.5

1

1.5N

MR

abs

orpt

ion

(m

V)

Ωv 5 0

Ωv 5 0.1 rad/s

Ωv 5 0.25 rad/s

Ωv 5 0.5 rad/s

T 5 0.40 Tc

P 5 29 bar

Ω 5 0

Ω 5 Ωv

5 0.9 rad/s

Larmor

r (cm)

z (c

m)

top NMR coil

bottom NMR

6 mm

H05 31.8 mT

H05 29.8 mT

f051.03 MHz

f050.965 MHz

Bottom platewith orifice21 0 1

1

2

3

4

5

6

7

8

9

10

11

0

3He-B

3He-B

3He-A

FIGURE 7 On the left, the NMR absorption spectra constitute an image of the‘flare-out’ order parameter texture in the long rotating cylinder (Kopu et al., 2000).The Larmor field, around which the NMR absorption is centred in the normal phase, ishere at 48.1mT. In the B phase, the Larmor value becomes a sharp edge beyond whichat higher fields the absorption vanishes, while most of the absorption is shifted tolower fields. The dominant absorption maximum on the left is the counterflow peak.Its height depends on the number of vortices N in the central cluster. N can bedefined in terms of the rotation velocity �v at which the vortices form theequilibrium state: N = Neq(�v). Here all counterflow peaks have been recorded at thecounterflow velocity � − �v = 0.8 rad/s. The conversion from peak height to N ingiven conditions (T ,�, P) can be obtained experimentally or from a numericalcalculation of the order parameter texture and the corresponding line shape (Kopu,2006). The two line shapes without a counterflow peak, but with a large truncatedmaximum bordering to the Larmor edge, are for the nonrotating sample (� = 0) andfor the equilibrium vortex state (N = Neq) at �v = 0.9 rad/s. For both of them, theabsorption at the site of the counterflow peak is close to zero. On the right, a NMRsetup is shown which was used to study the onset temperature of turbulence (cf. alsoFigure 13). Two different contours of the AB interface are shown when the A-phasebarrier layer is present at 0.6 Tc. The contours correspond to a current of 4 A (narrowA-phase sliver with curved concave interfaces) and 8A (wider A-phase layer with flatinterfaces) in the superconducting A-phase stabilisation solenoid (Finne et al., 2004b).

(Donnelly, 1991) (ρs is the density of the superfluid fraction)

fM = ρsκ s × (vL − vs), (2)

which acts on a vortex element s(ξ, t) with a tangent unit vector s =ds/dξ moving with velocity vL = ds/dt with respect to the superfluid


component, which locally has the velocity vs. The motion from the Magnusforce is damped by mutual friction which arises when the vortex moveswith respect to the surrounding cloud of normal excitations:

fmf = −γ0ρsκ s × [s × (vn − vL)] + γ ′0ρsκ s × (vn − vL). (3)

The mutual friction force has dissipative and reactive components, whichhere are expressed in terms of the two parameters γ0 and γ ′

0. Balanc-ing the two hydrodynamic forces, fM + fmf = 0, one gets the equation ofmotion for the vortex line element, which when expressed in terms of thesuperfluid counterflow velocity v = vn − vs has the form

vL = vs + αs × (vn − vs) − α′s × [s × (vn − vs)]. (4)

Here the dissipative and reactive mutual friction coefficients α and α′appear. Conversion formulae between different sets of friction parame-ters are listed by Donnelly (1991). Evidently solutions of the equation ofmotion can be classified according to the ratio of the two components.The important parameter proves to be ζ = (1 − α′)/α (Figure 1), which isthe equivalent of the Reynolds number of viscous fluid flow (Finne et al.,2003).

Two elementary examples are useful to consider. A single rectilinearvortex in rotating counterflow at v = �r moves such that its velocity com-ponents in the transverse plane consist of the radially oriented dissipativepart α�r and the azimuthally oriented reactive part −(1 − α′)�r, whenexpressed in the rotating coordinate frame. The former is responsible forthe contraction of the rectilinear vortices to a central cluster in Figures 5and 6. The latter causes the rectilinear vortex to rotate with the azimuthalflow with respect to the cylindrical wall.

The second simple consideration concerns the end point motion ofan evolving vortex along the cylindrical wall. Since the vortex end isperpendicular to the cylindrical wall, it has from Equation (4) a longi-tudinal velocity vLz = αv(�f, R, N) and an azimuthal component vLφ =−(1 − α′)v(�f, R, N). Evidently other parts of the vortex also contribute toits motion, in particular its curvature where it connects to the cylindricalwall. However, it turns out that the end point velocity is an approximateguide for the expansion of a single vortex in vortex-free rotation. Forcomparison, the calculated velocities of the two vortex ends in Figure 6 arevLz ≈ 0.84α�R ≈ 0.96αv(R)and vLφ ≈ 0.73(1 − α′)�R ≈ 0.83(1 − α′)v(R).The wave length of the spiral trajectory is λ = 2πR vLz/vLφ ≈ 5 mm andthe period p = 2πR/vLφ ≈ 50 s. Thus the end point motion can be usedto construct a simplified model of the motion of the two short vortices inFigure 6.


As seen in Figure 6, the spiral motion of the vortex end point along thecylindrical wall winds the rest of the evolving vortex around the centralvortex cluster with a wave vector Q such that QR = vLφ/vLz = ζ. The otheralmost straight end of the evolving vortex is fixed to a flat end plate ofthe cylinder and resides there at the edge of the vortex cluster, wherethe counterflow velocity is close to zero. Therefore, the helical twist isremoved only by a slow sliding of the vortex end along the end plate. Asseen in Figure 6, occasional reconnections between the twisted evolvingvortex and a straight outer vortex in the cluster or with a second oppositelytwisted vortex can help to reduce the twist. Finally, we see in Figure 6 thatwhile the evolving vortex is wound tightly around the cluster, this inducesKelvin-wave oscillations which propagate vertically along the vortices inthe cluster (Hänninen et al., 2007a).

2.3 Onset Temperature of Turbulence

When the dynamics calculated in Figures 5 and 6 is probed with mea-surements, the final state proves to depend crucially on temperature.At temperatures above the transition to turbulence, T > Tbulk

on , the cal-culations are confirmed and the number of vortices remains constant. Atlow temperatures, in contrast, evolving vortices may become unstable inapplied counterflow and trigger a turbulent burst. The evolution after theburst continues as illustrated in Figure 3. The final state is then consistentlyclose to the equilibrium vortex state. Interestingly, it turns out that for eachinitial configuration, such as those in Figures 5 or 6, there exists a specifictemperature Ton, which characterises the onset of turbulence: well aboveTon no turbulent burst occurs, while well below Ton a burst always occurs.

In these onset measurements, only two different types of final statesare observed: sufficiently far above Ton the number of vortices is con-served, while well below Ton close to the equilibrium number of vorticesis formed. The changeover takes place within a narrow temperature inter-val around Ton, typically within ±0.02 Tc. Within this interval, either of thetwo final states can emerge. The reason for the narrow width is the strong,nearly exponential temperature dependence of the mutual friction param-eter ζ = (1 − α′)/α, which controls the dynamic instability of seed vorticesevolving in the applied counterflow (Figure 1). As sketched in Figure 3,two sequential processes are needed to start the low-temperature evolu-tion: first, the single-vortex instability (Finne et al., 2006a), the precursoryprocess which becomes possible only at temperatures below Tbulk

on andwhich is responsible for generating a bunch of new evolving vortices. Sec-ond, the turbulent burst has to be triggered as a collective process in whichseveral evolving vortices interact and generate a sudden localised event ofturbulence, which expands across the entire cross-section of the rotatingcolumn but only over a short section of its length (of order ∼R).


The experimental confirmation of this scenario is obtained by examin-ing the final state as a function of temperature. By recording the line shapeof the NMR absorption profile, when the magnitude of the magnetic polar-isation field is swept across the resonance region, the number of vortices inthe final state can be determined. The line shapes of the two types of finalstates differ drastically, as seen in Figure 7, where all the spectra have beenrecorded at the same temperature of 0.40 Tc and where thus the integratedarea under each line shape is the same. The characteristic features are thelarge NMR shifts. These are controlled by the temperature and pressuredependent spin-orbit coupling. If the central vortex cluster is surroundedby applied counterflow at some sizeable velocity, a large sharp peak isformed, which is shifted downfield from the Larmor value. The numberof vortices in the central cluster can be determined from the height of thisso-called counterflow peak. At small vortex numbers (N � Neq), the reduc-tion in the counterflow peak height is directly proportional to the numberof rectilinear vortices N in the cluster.At larger N, the dependence becomesnonlinear and ultimately the peak height drops to zero well before Neq isreached, in practice around v(R) � 1 mm/s (Kopu, 2006). Accordingly, inthe equilibrium vortex state the line shape is radically different and easilydistinguished from a state with sizeable counterflow.

Ameasurement of Ton for any particular initial setup, such as in Figure 5for remanent vortices, requires repeating the measurement at differenttemperatures and recording the line shape in the final state. Surprisingly,it turns out that in the final state the vortex number is either preserved orit has increased close to that in the equilibrium vortex state. Practicallyno intermediate cases are observed. As a result, in practice, a measure-ment of Ton requires simply a visual check of the measured line shape inthe final state. This feature about the turbulent burst is similar to recentobservations from measurements with a closely spaced pair of vibratingwires in superfluid 4He (Goto et al., 2008). One of the wires is driven athigh oscillation amplitude as generator while the second is operated atlow amplitude as detector. Once turbulence has been switched on by run-ning the generator at high drive, the generator can be switched off andturbulent flow will still be maintained around the detector. Only if thedetector drive is reduced to sufficiently low level, turbulence ceases andthe flow around the detector returns to the laminar state. This shows thatonce turbulence has been switched on it can be sustained at much lowerflow velocities. Similarly, once the turbulent burst is started in the rotatingcolumn, turbulent vortex formation will continue until the counterflowvelocity has dropped close to zero in a section of the column of height ∼R.

The situation at temperatures above Ton is illustrated by the mea-surements on vortex remanence in Figure 8 (Solntsev et al., 2007). Thesemeasurements have been performed at two different temperatures aboveonset, where no increase in the number of vortices is expected. This is


0 100 200 300

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Bottom sample section

f=0.9 rad/s

t0

Ωf

Ω0

Δt

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Remnants evolve

Remnants decay

Ω Rotation driveEquil vortex state

FIGURE 8 Number of remanent vortices N (�t) for A-phase separated top andbottom sample sections, measured as a function of the annihilation time �t inthe temperature regime of laminar vortex motion. The results of these twoindependent measurements can be fitted in both cases with the solid curveN (�t) ≈ 2 · 103/(α�t + b) (�t in seconds, b ≈ 7 s). Inset: in the upper right corner,the sequence of rotations �(t) is shown which was used to perform themeasurement. The data are for 0.57 Tc with α = 0.60 and 0.7 Tc with α = 1.1.Parameters: �f = 0.9 rad/s, R = 3mm, length of top sample section ht = 44mm(41mm) and hb = 54mm (51mm) for the bottom section at 0.57 Tc (0.70 Tc).

confirmed by extracting from the counterflow peak height the number ofvortex lines in the final state. The measurement proceeds as sketched in theinset on the top right of Figure 8. An equilibrium vortex state is deceleratedto zero, and the vortices are allowed to annihilate for a time interval �tbefore rotation is turned back on. The measurement is repeated many timesby varying the annihilation time �t at zero rotation. The annihilation timeis found to govern the number of rectilinear vortex lines in the final stateand thus the number of remnants at the end of the annihilation period:N(�t) ∝ (1 + �t/τmf)

−1, where the mutual-friction-controlled time con-stant is τmf = [2α�0]−1 and �0 = �(t < 0). This is exactly as expectedfor the mutual-friction damped motion of vortices in the radial direction,when straight vortices move outward to annihilate on the cylindrical wallat zero rotation. For this to apply, the vortices have to be polarised alongthe cylindrical symmetry axis (Krusius et al., 1993). As seen in Figure 5(second from left, at t ≤ 600 s) this is the case: the polarisation remains athigh level even in the remanent state at zero rotation. Consequently, themeasurements in Figure 8 confirm that at constant temperature above Ton,the number of vortices in this experiment is controlled by the annihilationperiod �t and no uncontrolled increase occurs.

It is useful to note some additional features about vortex remanence inthe measurements of Figure 8. Let us denote the number of remnants afterthe annihilation period with N (�t) and the number of rectilinear lines in


the final state with N. Although the annihilation time �t controls the num-ber of remnants N (�t), the result N = N is independent of �t. It is also toa large extent independent of how the measurement is performed, that is,what the rotation velocity �f in the final state is or what acceleration � isemployed to reach �f (as long as N (�t) < Neq(�f) or the critical velocityfor vortex formation, �f − �v < vc/R, is not exceeded). Furthermore, theresult N = N is established separately both for the top and bottom sectionsof the cylinder, when these are separated by a magnetic-field-stabilisedA-phase barrier layer (cf. Figure 7), and for the entire cylinder withoutA-phase barrier. The A-phase barrier layer prevents vortices from travers-ing across the AB interfaces at low counterflow velocity (Blaauwgeerset al., 2002). In this way, the number of remanent vortices N (�t) hasbeen found to be proportional to the length h of the cylinder (as longas h � R). All these properties are consistent with the conclusions thatwhen T > Ton, the vortex number is conserved in dynamical processes, theannihilation decay of remnants is a laminar process regulated by mutualfriction damping, and that pinning is weak. For simplicity, we neglect vor-tex pinning altogether and assume ideal wall properties throughout thisreview.

The situation at temperatures around Ton is illustrated by the mea-surements in Figure 9 which determine Ton for this particular choice ofinitial state (de Graaf et al., 2007; Solntsev et al., 2007). The probabilityof the turbulent burst is plotted as a function of temperature, when theannihilation time �t = 20 min (on the left) and �t = 2 min (on the right).The striking feature is the abrupt changeover from the laminar behaviour,where the vortex number is conserved, to turbulence, where the vortexnumber surges close to Neq and the system relaxes to its minimum energystate. The centre of the narrow transition defines the onset temperature ofturbulence Ton, which proves to be different for the two cases studied inFigure 9.

As seen in Figure 9, Ton depends on the annihilation time �t and thuson the initial number and configuration of evolving remnants N (�t) atthe moment when the step increase in rotation from zero to �f is applied.Calculating from the results in Figure 8, one finds that the number ofremnants at the start of acceleration is N i ≈ 40, when �t = 2 min and T ≈Ton = 0.44 Tc, and Ni ≈ 10, when �t = 20 min and T = 0.39 Tc. Thus, at ahigher temperature, a larger number of remnants is needed to achieve theturbulent burst. Both panels in Figure 9 refer to the top sample half where,with no orifice, there is no preferred site for the remnants to accumulate,and the turbulent burst occurs randomly at any height z in the column(de Graaf et al., 2007). Similar measurements at different values of �t andfinal rotation velocity �f show that the onset temperature depends weaklyon both the initial number of remnants Ni and the applied flow velocity.


Top sample section

0.35 0.4 0.45 0.5 0.550

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σT5 0.02 Tc

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Vortex numberconserved

Alwaysturbulence

Vortex numberconserved

Alwaysturbulence

FIGURE 9 Measurements on the onset temperature Ton of the transition toturbulence. The measurements are performed similar to those in Figure 8 and startfrom an initial state which is obtained by decelerating an equilibrium vortex state at1.7 rad/s to zero at a rate 0.01 rad/s2. The remaining vortices are left to annihilate fora period �t at � = 0. Rotation is then increased to �f at a rate 0.02 rad/s2. When alltransients have decayed, the number of vortices is measured in the final steady stateat �f. The result is plotted as a function of temperature with 30–40 data points perpanel. The solid curve is a gaussian fit which represents the probability for turbulencewith a half width σT = 0.02 Tc, centred around Ton. Comparing results in the twopanels for �t = 20min and 2min, we see that Ton decreases with increasing �t sincethe number, average size and density of remnants are reduced as �t increases. Bothpanels have been measured for the upper sample section which is separated from thebottom half with an A-phase barrier layer. Parameters: R = 3mm, h = 45mm andP = 29.0 bar.

These dependences can be summarised in the form

ζ(Ton)−1 ∝ N 0.3

i �1.3f . (5)

Thus the onset temperature Ton depends primarily on the mutual fric-tion parameter ζ(T), but also weakly on other factors which influencethe likelihood of achieving locally somewhere in the maximum availablecounterflow velocity a density of evolving vortices which allows to triggerthe turbulent burst. Among these additional factors, most important are(i) the applied counterflow velocity v = vn − vs, (ii) the number and con-figuration of the injected seed vortices and (iii) the sample geometry. InFigures 8 and 9, we examined the response of remanent vortices to a step-like increase in rotation. The same measurements can also be performedby starting from the equilibrium vortex state at finite rotation, as discussedin the context of Figure 6. In fact, the most extensive study of the scalinglaw in Equation (5) was performed using this approach.


Finally we note that in the rotating container, all measured onset tem-peratures, which depend on the presence of the precursor, are found to bebelow the transition to turbulence in the bulk: Ton < Tbulk

on . Furthermore,since the onset also depends on the applied counterflow velocity in Equa-tion (5), the instability is expected to occur first close to the cylinder wall,where the applied velocity v = � r reaches its maximum value at r = R.Thus the reconnection of the expanding loop will most likely occur withthe wall. Surprisingly, it is also found that once the instability is triggered,the turbulent burst essentially always follows next, since little if no increasein the vortex number is detected at T ∼ Ton in such cases where the turbu-lent burst does not switch on (cf. Figure 9). To provide more understandingon the role of the single-vortex instability as the precursor mechanism toturbulence, we next examine it in the onset temperature regime, T ∼ Ton,where the instability proceeds sufficiently slowly in time so that it can bemonitored with continuous-wave NMR measurement.

2.4 Single-Vortex Instability in Applied Flow

Since the time when it was first understood that superfluid turbulenceis made up of tangled quantised vortices (Hall and Vinen, 1956; Vinen,1961), the most basic question has been its onset as a function of appliedcounterflow velocity: how is turbulence started and what defines its criticalvelocity? An important clue was provided by the rotating experiments ofCheng et al. (1973) and Swanson et al. (1983), who found that rectilinearvortex lines in rotation are broken up in a turbulent tangle if a heat currentis applied parallel to the rotation axis. The thermal current is transported asa counterflow of the normal and superfluid components along the rotationaxis. Rectilinear vortices become unstable in this parallel flow and abovea low critical velocity transform to a tangle which tends to be aligned inthe plane transverse to the heat current.

This phenomenon was explained by Glaberson et al. (1974) whoshowed that an array of rectilinear vortices becomes unstable in longitudi-nal counterflow above the critical velocity v = 2(2�κe)

1/2, where κe ≈ κ isan effective circulation quantum (supplemented with the logarithmic cut-off term κe = (κ/4π) ln (�/a0), where the average inter-vortex distance is� ∼ (κ/2�)1/2 and the vortex core radius a0). The instability appears whena Kelvin-wave mode with wave vector k = (2�/κe)

1/2 starts to build up,whose amplitude then grows exponentially in time. The Glaberson insta-bility has also been examined in numerical calculations which qualitativelyconfirm the instability and the vortex tangle in the transverse plane whichstarts to form above a first critical velocity (Tsubota et al., 2003).

In general, the dispersion relation of a helical Kelvin-wave disturbance∝ exp[−i(ωkt − kz)] can be written as (Donnelly, 1991; Finne et al., 2006b)

ωk(k) = κek2 − α′(κek2 − kv) − iα(κek2 − kv). (6)


In the absence of flow (v = 0), these Kelvin modes are always damped,at high temperatures they are actually overdamped, but at low temper-atures (α < 1 − α′) this is not the case. In applied flow (v > 0), the longwave length modes with 0 < k < v/κe become exponentially unstable. Ifan evolving vortex accumulates enough length L‖ parallel to the appliedflow, then a disturbance with wave length λmin ∼ L‖ ∼ κe/v ∼ 1/kmax maystart to grow. The expanding loop may reconnect, either with the wall of thecontainer, with itself, or with another vortex. This leads to a growing num-ber and density of evolving vortices, which ultimately start interacting andtrigger the onset of turbulence in the bulk.

No rigorous analytical calculation has been presented of the single-vortex instability in the rotating container, but a simple scaling modelillustrates the problem. Consider a vortex ring in vortex-free counter-flow, which is initially perpendicular to the plane of the ring in a rotatingcylinder of radius R. If the ring is large enough, then it expands untilit reaches the container size R. The time needed for this expansion is oforder δt ∼ R/αv, where v is the average normal velocity through the ring.The ring also has a self-induced velocity component vr ∼ κe/R, whicharises from its own curvature and is directed along the normal of theplane of the ring. Because of this velocity component, the plane of the ringis rotated away from being perpendicular to the azimuthal flow in thecylinder while it drifts in the flow. During the time δt, the vortex lengthparallel to the flow becomes of order (1 − α′)vr δt. Equating this to L‖, itis seen that the instability condition L‖ � λmin leads to the requirementζ = (1 − α′)/α � 1. This condition is virtually independent of velocity; theonly restriction is imposed by the finite container radius, L‖ < R, whichdefines a critical velocity vc ∼ κe/R. Typically, the time spent by a vor-tex in radial motion before reaching the sample boundary is of orderR/(α�R) = (α�)−1. At 0.45 Tc and � = 0.6 rad/s, the inverse of this quan-tity equals 0.20 s−1, which fits with the measured vortex generation rateof dN/dt = N = 0.23 s−1 in Figure 10. However, numerical calculationsconfirm that the presence of surfaces is required to demonstrate the single-vortex instability in usual experimentally relevant flow conditions (Finneet al., 2006a). Also the calculations demonstrate that the instability is notcharacterized by a unique critical velocity, since it depends on the rela-tive orientation of the flow with respect to the vortex, while the vortexexpands in helical motion in the rotating cylinder. Thus the above modelis incomplete.

Let us now examine direct observations of the single-vortex instabilityin rotating flow. In the onset temperature regime, T ∼ Ton, in about halfof the measured cases, which lead to a turbulent burst, a slow increasein the number of vortex lines N(t) can be observed to precede the turbu-lent burst. If present, the increase is invariably followed by a turbulentburst. Thus it appears reasonable to associate the slow increase with thesingle-vortex instability. In Figure 10, the number of vortex lines N(t) is


2100 0 100 200 300Time t (s)

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FIGURE 10 Experimental illustration of single-vortex instability as precursor of bulkturbulence. The number of vortices N(t) is recorded with NMR coils at the top andbottom ends of the sample. As seen in the inset, initially the sample is in theequilibrium vortex state at �i = 0.05 rad/s with N ≈ 30 vortices, of which close toone half connect to the cylindrical sidewall. Rotation is then increased to a new stablevalue �f = 0.6 rad/s, which is reached at t = 0. During the ramp to �f, thecounterflow builds up, compresses the rectilinear sections of all vortices to a centralcluster, and starts the spiral motion of the vortex ends connecting to the sidewall.Eventually in the increased applied flow at �f, the instability starts to generate newvortices which contribute to the average slow rate N of vortex formation, shown bythe solid straight line. After about 140 s, the turbulent burst occurs 63mm above thebottom end plate. Vortex fronts traveling up and down along the column thenapproach the two detector coils and reach their closer ends as indicated by verticalarrows (at 230 s and 275 s). The filled data symbols are derived from experimentallycalibrated counterflow peak heights and the open symbols from order parametertexture calculations fitted to the NMR signal in the non-rotating state.

plotted as a function of time while the precursor generates new vorticesat slow rate. In this example, the increase in N(t) is almost linear (solidline) until the turbulent burst sets in and starts the vortex front motionalong the rotating column both upwards and downwards from the siteof turbulence. At �f = 0.6 rad/s, the slow, increase lasts in this examplefor about 140 s, generating approximately one vortex every five seconds,until some 30 new vortices have been created and the turbulent burstmanages to switch on. The time interval from t = 0 to the turbulent burst


is called the burst time which here is tb = 140 s. At larger �f, the burst timeis shorter in duration; for example, in a repetition of the measurements inFigure 10 at �f = 1 rad/s, the turbulent burst was found to start in lessthan 30 s.

Two further observations about the precursor can be made fromFigure 10. First, vortex formation proceeds independently in different partsof the sample. At �f = 0.6 rad/s, it takes more than 300 s for a vortex cre-ated at one end of the sample to reach the other end. Still, vortex formationat the top and bottom ends is observed to proceed at roughly the same rate.Thus, vortex generation by the single-vortex instability is not localised,in contrast to the turbulent burst. The random occurrence of the single-vortex instability agrees with the notion of ideal walls (or at least weakpinning), as opposed to a vortex mill localised at a surface defect on thecylinder wall.

Secondly, in Figure 10 the equilibrium vortex state at low initial rota-tion �i = 0.05 rad/s has been used to introduce evolving vortices in theapplied flow. This approach provides a more reproducible initial vortexconfiguration than remanent vortices since the number of those vortices,which connect to the cylindrical sidewall, is primarily determined bythe misalignment between the cylinder and rotation axes (Figure 6). Ina given experiment, the residual angle between the two axes is generallya constant.

To appreciate the influence of the vortices curving to the sidewall, theexperiment was repeated differently. A cluster with only rectilinear vor-tices (N < Neq) was prepared at higher temperatures and was then cooledbelow 0.5 Tc. As long as this cluster is separated by a sufficiently widevortex-free counterflow annulus from the cylindrical boundary, � can beincreased or decreased without change in N at any temperature down to0.35 Tc (which is the lower limit of the so far measured onset tempera-tures Ton). If � is reduced too much, the cluster makes contact with thecylindrical sidewall, some outermost vortices become curved, and duringa subsequent increase of �, while T < 0.5 Tc, the behaviour in Figure 10is reproduced. Therefore, we are led to assume that, to observe the vortexinstability, at least one curved vortex connecting to the cylindrical sidewallneeds to be present. At temperatures below 0.35 Tc, this may not be thecase, since in rapid changes of rotation even rectilinear vortices seem to bedestabilised (Figure 4).

More statistics on the properties of the precursor have been collectedfrom measurements similar to that in Figure 10 by de Graaf et al. (2007).Important characteristics are the initial rate of vortex generation N(t = 0)and the burst time tb. These can be examined for events with sufficientlylong burst times tb � 20 s, so that the rate of the counterflow peak heightdecrease with time can be adequately resolved. In general, it is found that Nincreases and tb decreases rapidly with decreasing temperature below Ton.


To find events with well-resolved preturbulent vortex generation and longburst time, one has to scan for data (i) in the onset temperature regime, T ≈Ton, (ii) with a low initial formation rate N � 1 vortex/s, (iii) by startingfrom a state with a small number of seed vortices and (iv) at low appliedflow velocity.

These measurements demonstrate that the precursor generates newindependent vortex loops which start to evolve along spiral trajectoriestowards the final state of a rectilinear vortex line. When the density ofevolving vortices rises sufficiently, so that interactions between them inthe bulk volume become possible, then the process is terminated in aturbulent burst. The burst is a localised event which from one measure-ment to the next happens randomly at different heights z of the sample(Figure 7). The measured properties of the precursor are consistent withthose expected for a single-vortex instability based on the excitation ofKelvin-wave modes of sufficiently long wavelength. Overall, measure-ments in the onset regime reveal the precursor mechanism, owing to thestrongly temperature dependent mutual friction of 3He-B, which makesthe precursor observable within a narrow temperature interval around theonset temperature. At lower temperatures, the turbulent burst developsso rapidly that the measuring techniques, which have been employed sofar, are not fast enough to capture the details. The latter case is the typicalsituation in superfluid 4He experiments.

2.5 Numerical Calculation of Dynamic Vortex Generation

Numerical calculations on vortex dynamics are carried out with thevortex filament model introduced by Schwarz (1988). With today’s com-puting power, one uses Biot-Savart integration along all vortex linesso that the superfluid velocity field from vortices is obtained from(Hänninen et al., 2005)

vs,ω(r, t) = κ

4π

∫(s − r) × ds

|s − r|3 . (7)

The line integral is taken along all vortices in the system, s(ξ, t) denotes thelocation of the vortex core at time t and ξ is measured along the arc lengthof the vortex core. In the presence of solid boundaries, the total super-fluid velocity field, vs = vs,ω + vb, is modified by the boundary-inducedvelocity vb. At a plane boundary, one can use image vortices to satisfy therequirement of zero flow through the boundary, n · vs = 0, where n is theunit vector along the surface normal. More generally, we obtain vb = ∇�

by solving the Laplace equation ∇2� = 0 combined with the requirementthat at the boundary n · ∇� = −n · vs,ω.


No surface pinning or even surface friction is generally included, theboundaries are assumed ideal, as indicated so far by measurements on3He-B in smooth-walled simple cylindrical containers. Mutual friction inthe bulk superfluid is included using the equation of motion (Equation (4))for the vortex element at s(ξ, t), which moves with the velocity vL = ds/dt.For the mutual friction parameters α(T, P) and α′(T, P), one uses the 3He-Bdata measured by Bevan et al. (1997) at 10 and 29 bar pressures. A recon-nection between two vortex segments is enforced if they have driftedwithin a distance from each other, which is less than the minimum spatialresolution of the calculation (usually ∼0.05 mm). The configuration afterreconnection should correspond to shorter overall vortex length than theinitial state. In practice, the computing time limits severely what can becalculated and what becomes too time consuming. Therefore, the practicalimplementation becomes of great importance, how the Biot-Savart inte-gration and the proper solution for the boundary conditions are workedout. For details, we refer to de Graaf et al. (2007).

In Figure 11, two snapshots are shown from calculations on vortexformation and the configurations which evolve in a rotating cylinder (deGraaf et al., 2007). Recently formed younger vortices are here in helicalconfigurations on the outer circumference closer to the cylindrical wall.There in the outer regions, one can see loops of Kelvin-waves, small sep-arated loops with both ends of the vortex on the cylindrical wall, andeven closed vortex rings (lower right corner at t = 50 s). Since it is pri-marily surface reconnections at the cylindrical wall, which contribute tothe formation of new vortices in the early stages of the calculation (att < 100 s), the many newly formed short loops are still close to the sidewall.

t 5 77.5 st 5 50.5 s

FIGURE 11 Two snapshots from a calculation of vortex generation in a rotatingcylinder. The summary of these calculations with results accumulated over more than100 s is shown in Figure 12.


Further inside the cluster, one can see older and straighter vortices whichcongregate within the central parts.

The general observation from these calculations is that evolving vor-tices in a rotating sample are more stable in the numerical experimentthan in measurements. For instance, in Figure 11 vortex formation has tobe started from an artificial initial configuration (Finne et al., 2006a). Thisconsists from an initial single vortex ring which is placed in the plane per-pendicular to the rotation axis at height 0.2 h slightly off centre, to breakcylindrical symmetry (de Graaf et al., 2007). This is an unstable configura-tion where Kelvin-waves of large amplitude immediately form and thenreconnect at the cylindrical wall. The end result is the sudden formation ofroughly 30 vortices which have one end on the bottom end plate and theother moving in spiral trajectory along the cylindrical wall. After the initialburst, the later evolution is followed as a function of time t, the number ofvortices N(t) is listed and the reconnections of different type are classified.The results are shown in Figure 12.

In Figure 12, one keeps account of all reconnection processes whichoccur in the rotating sample as a function of time while it is evolvingtowards its final stable state with an array of rectilinear vortices andN → Neq. After the initial burst of the first ∼30 vortices, N increasesfirst gradually, but after about 50 s, the rate N picks up. During the first50 s, reconnections in the bulk do not contribute to the generation of newvortices, but later such processes also start to appear. However, even dur-ing the later phase, a reconnection of a single vortex at the cylindricalwall, while Kelvin-waves expand along this vortex, remains the dominantmechanism of vortex generation. This is seen from the fact that the curve forN follows closely that of the successful surface reconnections (dashed curvemarked as ‘�N = +1’). The most frequent reconnections after the first 40 sare denoted by the solid ‘�N = 0’ curve and occur in the bulk between twodifferent vortices. These inter-vortex reconnections do not lead to changesin N and are primarily associated with processes occurring between thetwisted vortices in the bundle further away from the wall.

The inset in Figure 12 compares the rates of vortex generation fromreconnections at the wall and in the bulk. The reconnection of a singlevortex at the cylindrical wall is clearly the most important mechanism forthe generation of new independent vortex loops in the early stages of thecalculation. The dominant role of such wall reconnections is compelling.A second important consideration is correspondence with measurement.The obvious difference is the higher stability of evolving vortices in thecalculation as compared to experiment. In Figure 12, the rate of vortexgeneration remains modest, no clearly identifiable turbulent burst can bedistinguished, and the vortex number approaches the equilibrium valuefrom below. After 115 s of evolution, the vortex number has progressed toN ≈ 400 , where the increase is almost stopped, well below the saturation


0 20 40 60 80 1000

50

100

150

200

250

300

350

400

450

Time (s)

Cum

ulat

ive

num

ber

of r

econ

nect

ions

& v

ortic

es

N

ΔN 511 Wall

0

21

ΔN 5 0

11

Bulk

0 20 40 60 80 100

0

1

2

3

4

5

6

Bulk

Wall

Time (s)

Vor

tex

form

atio

n ra

te (

1/s)

Startup

Removed

FIGURE 12 Calculation of the cumulative number of reconnections and vortices in arotating cylinder. The different curves denote: (�N = 0, solid curve) reconnections inthe bulk which do not change N; (+1, dashed) reconnections with the cylindrical wallwhich add one new vortex loop; (N, solid) total number of vortices; (removed,dash-dotted) small loops which form in reconnections mainly close to the cylindricalwall, but which are contracting and are therefore removed; (+1, solid) reconnections inthe bulk which add one vortex and (−1, solid) which remove one vortex; (0, dashed)reconnections at the cylindrical wall which do not change N. Inset: averaged rate ofincrease in N owing to reconnections on the cylindrical wall and in the bulk. The largeinitial peak in the boundary rate represents the starting burst, which is used to startvortex formation. Parameters: R = 3mm, h = 10mm, � = 0.9 rad/s and T = 0.35 Tc(where α = 0.095 and α′ = 0.082).

value of Neq ≈ 780 (de Graaf et al., 2007). This is the general experiencefrom calculations on an ideal rotating cylinder, with smooth surfaces andno surface friction or pinning. The calculations become more and moretime consuming with decreasing temperature, which limits the possibil-ities to obtain a more comprehensive understanding of their predictionsand of the origin of the differences with measurement. The low probabilityof the single-vortex instability in the calculations appears to be a particularproperty of rotating flow in a circular cylinder since linear pipe flow, forinstance, displays a steady rate of vortex generation (de Graaf et al., 2007).

Clearly numerical calculations provide important illustrations andguidance in situations where measurements answer only specific lim-ited questions. The calculations take full account of interactions betweenvortices and between a vortex and the ideal container wall. Nevertheless,


the correspondence between calculation and measurement is not satisfac-tory at present, when we speak about the single-vortex instability and theonset of turbulence in a rotating cylinder. It appears that some mechanism,which makes vortices more unstable and adds to the vortex generationrate, is missing from the calculations. The difficulty is likely to reside onthe cylindrical wall, where the assumption of ideal conditions should beexamined closer. Attempts in this direction have so far not produced moreclarification. However, these uncertainties about the mechanisms behindthe single-vortex instability in rotating flow do not change the fact that, atlow vortex density Kelvin-wave formation on a single vortex, followed bya reconnection at the surface, is the only efficient mechanism for generatingnew vortices.

2.6 Summary: Onset of Turbulence

Since the advent of 3He-B, new possibilities have appeared to study tur-bulence. First, it has become possible to distinguish and characterise, inmeasurements with large samples, vortex formation at a stable repro-ducible critical velocity, vortex remanence and turbulent proliferation ofvortices. Second, the mutual friction dissipation α(T) with strong temper-ature dependence around α ∼ 1 has made it possible to evaluate the role ofmutual friction in the onset of turbulence. The important dynamic param-eter proves to be ζ = (1 − α′)/α. It controls the onset of the single-vortexinstability, where an evolving vortex becomes unstable and generates, dur-ing a reconnection at the wall, a new vortex loop. After several such events,the density of evolving vortices is sufficient to produce a turbulent burst.The necessary condition is ζ � 1 to start the cascade of the single-vortexinstability followed by the turbulent burst.

The single-vortex instability becomes possible only at temperaturesbelow the turbulent transition in the bulk volume and thus Ton ≤ Tbulk

on .The onset temperature Ton of these two series-coupled processes has beenfound to obey a power-law dependence which relates the mutual frictionparameter ζ to the magnitude of the ‘flow perturbations’ in Equation (5).Well above Ton, no new vortices are detected (with a resolution < 10 newvortices), while well below Ton all final states are found to be equilibriumvortex states with close to the equilibrium number of vortices, N � Neq.In the onset regime itself, T ∼ Ton, one finds events with and withoutturbulent burst but surprisingly practically no incomplete transitions withNi < N � Neq.

In the intermediate temperature regime 0.3 Tc < T < 0.6 Tc, the equi-librium vortex state is reached after a single turbulent burst. In fact, in themeasurements with the sample setup of Figure 7, no case of two or morealmost simultaneous bursts was identified above 0.35 Tc. Apparently theprobability of the single vortex instability to start a turbulent burst is still


low at these temperatures. Second, after the burst, the vortex front movesrapidly and removes the vortex-free flow. At these intermediate tempera-tures, the burst is both spatially and temporally a localised event in a shortsection (of length ∼R) of the column. From one measurement to the next,it occurs randomly at different heights of the column. Below 0.3 Tc, thelongitudinal propagation velocity of vortices becomes slow and evolvingvortices go rapidly unstable everywhere. As a result turbulence tends tobe both spatially and temporally more extended, filling larger sections ofthe column. The later events, the evolution after the turbulent burst, arethe subject of the next section.

3. PROPAGATING VORTEX FRONT IN ROTATING FLOW

3.1 Introduction

In rotation at constant angular velocity, the steady state superfluidresponse is generally not turbulent. Nevertheless, transient states ofturbulence can be formed by rapidly changing the rotation velocity, espe-cially if the sample container does not have circular cross-section or itssymmetry axis is inclined by a larger angle from the rotation axis. Thedecay of turbulence and the approach to equilibrium can then be mon-itored at constant �. The normal component relaxes back to solid bodyrotation by means of viscous interactions, while the superfluid compo-nent adjusts much slower, coupled only by mutual friction dissipationfrom vortex motion with respect to the normal component and (if any) bythe deviations of the container walls from being axially symmetric aroundthe rotation axis. Such measurements on transient turbulence are gener-ally known as spin-up or spin-down of the superfluid component. Thisused to be an important topic in superfluid 4He work in the fifties and six-ties (Andronikashvili and Mamaladze, 1967) but was replaced (with fewexceptions (Adams et al., 1985)) by other methods which were expectedto lead to more straightforward interpretation.

The turbulent burst, which suddenly starts the motion of N ≈ Neq vor-tices along the rotating column at temperatures T � Ton, as discussed inSection 2 (cf. Figure 3), provides a novel technique to investigate transientturbulence in rotation. Originally it was assumed that this motion wouldtake place as a tangle of vortices, which spreads longitudinally along therotating column. It was soon realised from NMR measurements (Eltsovet al., 2006b) that this could not be the case; rather the propagating vor-tices were highly polarised and had to be coiled in a helical configurationowing to their spirally winding motion. This recognition presented a newproblem: is there any room at all for turbulence in this kind of motion and ifthere is, how is it expressed? Or perhaps the nature of the motion changes


on approaching the zero temperature limit, when mutual friction dissi-pation vanishes α ∝ exp(−�/T)? These questions provided the incentiveto examine the propagation more closely and to measure its velocity asa function of temperature. The results demonstrate that turbulent lossesdepend crucially on the type of flow, flow geometry, external conditions,the physical properties of the superfluid itself, etc.

A measurement of the front propagation in the laminar and turbulenttemperature regimes allows one to determine the rate of kinetic energydissipation. The measurement proceeds as follows: the initial startingstate is the rotating vortex-free state, the so-called Landau state, whichis metastable with much larger free energy than the stable equilibriumvortex state. The latter consists of rigidly co-rotating normal and super-fluid components, owing to the presence of a regular array of rectilinearvortices, while in the vortex-free state the superfluid component is notrotating at all: it is at rest in the laboratory frame of reference. When theturbulent burst is triggered in the Landau state, a rapid evolution towardsthe equilibrium vortex state is started, where a boundary between thevortex-free and the vortex states propagates along the rotating columnand displaces the metastable vortex-free counterflow. Particularly at tem-peratures below 0.4 Tc, the boundary has the form of a sharp thin vortexfront which travels at a steady velocity Vf. The dissipation rate of the totalkinetic energy, E(t), is related to Vf as

dE/dt = −πρsVf �2R4/4 . (8)

By measuring Vf, one determines directly the energy dissipation dE/dt asa function of temperature.

At high temperatures, the motion is laminar and the front velocity isdetermined by mutual friction dissipation between the normal and super-fluid components, Vf(T) ≈ α(T)�R. Below 0.4 Tc, Vf(T) deviates moreand more above the laminar extrapolation (Eltsov et al., 2007), in otherwords, the dissipation becomes larger than expected from mutual frictionin a laminar flow. At the very lowest temperatures, a striking anomalybecomes apparent: dE(t)

/dt does not go to zero, but the measured veloc-

ity Vf(T) appears to level off at a constant value which corresponds toan effective friction αeff ∼ 0.1, even though α(T) → 0, when T → 0. Evi-dence for a similar conclusion has been offered by the Lancaster group(Bradley et al., 2006), who measured the density of the vortex tangle cre-ated by an oscillating grid and found that this kind of turbulence decaysat a temperature-independent finite rate below 0.2 Tc.

When mutual friction decreases and turbulent motions in the vortexfront cascade downward to progressively smaller length scales, eventu-ally individual quantised vortex lines must become important. This isthe quantum regime of superfluid hydrodynamics. The energy cascade


on length scales smaller than the inter-vortex distance and the nature ofdissipation on these scales are currently central questions in superfluid tur-bulence (Vinen and Niemela, 2002). Theoretical predictions exist on the roleof nonlinear interactions of Kelvin-waves and the resulting Kelvin-wavecascade, which is ultimately terminated in quasi-particle emission (Kozikand Svistunov, 2004, 2005a, 2008a; Vinen, 2000; Vinen et al., 2003), or on theimportance of reconnections which could rapidly redistribute energy overa range of scales and also lead to dissipation (Svistunov, 1995). From theirfront propagation measurements, Eltsov et al. (2007) conclude that theKelvin-wave cascade accounts for an important part in the increased dis-sipation below 0.3 Tc. The different sources of dissipation in this analysisare discussed in Sections 3.4 and 3.5.

It is worth noting that a propagating turbulent vortex front has manyinteresting analogues in physics (van Saarloos 2003). For instance, it issimilar to the propagation of a flame front in premixed fuel. Flame frontpropagation can also proceed in laminar or in turbulent regimes. In thelatter case, the effective area of the front increases and its propagationspeed becomes higher than in the laminar regime. This property finds itspractical use in combustion engines but has also been used by Blinnikovet al. (2006) to describe intensity curves of type Ia supernovae. In all suchcases, a metastable state of matter is converted to stable state in the frontand Vf is determined by the rate of dissipation of the released energy.

3.2 Measurement of Vortex Front Propagation

The velocity of the vortex front was measured by Eltsov et al. (2007) withthe setup in Figure 13. The initial vortex-free state was prepared by warm-ing the sample above 0.7 Tc, where remanent vortices annihilate rapidly,and by then cooling it in the vortex-free state at constant rotation to thetarget temperature. Two different procedures were used to trigger the tur-bulent burst at the target temperature. These are sketched in Figure 13. Inboth cases, the front velocity is determined by dividing the flight distanceby the flight time, assuming that the front propagates in steady-state con-figuration. Although this is not exactly true, for instance owing to initialequilibration processes which follow injection, it is assumed for now thatthis simplification is justified. Especially, since the two injection techniquesfor different propagation lengths give the same result.

The first injection technique (depicted on the left, Figure 13) makesuse of remanent vortices (Solntsev et al., 2007). By trial and error it wasfound that one or more remnants can be freed with a small step increasein rotation from the region around the orifice on the bottom of the samplecylinder. This was done by increasing � in small steps, until at some pointusually above 1 rad/s a remnant starts expanding which below 0.35 Tcimmediately gives rise to a turbulent burst. The ensuing vortex front then


to heatexchanger

1.5 cm

0.5 cm

Top NMR coilH0 � 31.8 mTf0 � 1.03 MHz

H0 � 29.8 mTf0 � 0.965 MHz

bottom NMR coil

Injectionof vortices

Ø 6 mm

1.0 cm

Vortex injection:from AB interface

Vortex injection:from orifice

11 cm

Injectionof vortices

3He-Bvortex-free

3He-A equilvortex state

3He-Bvortex-free

Bottom platewith orifice

3He-B withvortices

Ω � ΩcAB0.8 ... 1.6 rad/s

FIGURE 13 Experimental setup for measuring the propagation velocity of the vortexfront in the rotating column. Two methods are shown for measuring the front motionacross different flight lengths in one single experimental setup. On the left, the seedvortices are tiny remnants at the orifice. In increasing rotation at � � 1 rad/s, theyproduce a turbulent burst in the volume around the orifice below the bottomdetector coil. A single vortex front is then observed to pass first through the bottomcoil and later through the top coil. The time difference separating the signals from thepassing front over the flight path of 90mm defines the front velocity Vf. On the right,the seed vortices are injected via the Kelvin-Helmholtz shear flow instability of thetwo AB interfaces. The injection event is followed instantaneously by a turbulentburst close to the AB interface on the B-phase side. A vortex front is then observed topropagate independently both up and down along the cylinder. The lengths of theflight paths are equal for the upper and lower halves. In Figure 14, it is explained howthe flight time is determined in this case.

propagates upwards along the entire column through both pick-up coilsin succession.

The second injection method (depicted on the right, Figure 13) relies onthe superfluid Kelvin-Helmholtz (KH) instability of the interface betweenthe A and B phases of superfluid 3He (Blaauwgeers et al., 2002). Two stableAB interfaces are formed by applying a specially configured magnetic fieldwhich stabilises a narrow A-phase barrier layer over the midsection of thesample cylinder. The shear flow instability of these two AB interfaces is


controlled by rotation velocity, temperature and the stabilisation field. Atthe target temperature, the instability can be triggered with a step increaseof the rotation velocity or of the stabilisation field (Finne et al., 2004b).The instability causes vortices from the A phase to escape across the ABinterface into the vortex-free B-phase flow in the form of a bunch of smallclosely packed loops. Once in the B phase, at temperatures below 0.59 Tc,the loops immediately interact and generate a turbulent burst. Two vortexfronts then propagate independently up and down from the AB interfaces,arriving to the top and bottom pick-up coils practically simultaneously(since the setup is symmetric with respect to the midplane of the stabilisa-tion field). An example of the NMR readout as a function of time is shownin Figure 14. The KH shear flow instability and the associated vortex leakacross the AB interface have been extensively described in the review byFinne et al. (2006b).

In Figure 14 one of the signal traces records the absorption at the coun-terflow peak (cf. Figure 7). It is at maximum in the initial vortex-free flowat v = vnφ − vsφ (where vnφ = � r and vsφ = 0 in the laboratory frame).To trigger the KH instability, � is increased by a small increment acrossthe critical rotation velocity �cAB, which is instantaneously registered asa small increase in absorption level, owing to the increased counterflowvelocity. Following the instability and the turbulent burst, the vortices sub-sequently propagate along the column, but a response in the counterflowpeak height is not observed until they reach the closer end of the NMR coil.From thereon, the absorption in the peak rapidly decreases and drops tozero. Keeping in mind that the peak height measures the azimuthally cir-culating flow in the transverse plane (cf. Equation (1)), its sudden removalrequires that the passage of the vortices through the coil must occur asan organised sharp front, followed by a highly polarised state behind thefront. The passage is characterised by the time τCF, which is defined inFigure 14. The propagation velocity Vf of the front is determined from itsflight time, measured from the AB interface instability (when�(t) = �cAB)to the arrival of the front at the closer edge of the pick-up coil, that is,where the rapid drop in the counterflow peak height starts. Compared tothe flight time, the AB interface instability and the turbulent burst can beconsidered as instantaneous.

The second signal trace in Figure 14 records the absorption in the Lar-mor peak. It is near zero in the initial vortex-free state, displays a sharpmaximum after the collapse of the counterflow peak, and then decays to asmall but finite value, which is a characteristic of the final equilibrium vor-tex state. The transient maximum is the new feature, which arises from flowin the axial direction, created by a helically twisted vortex bundle (Eltsovet al., 2006b). With increasing wave vector Q of the helix, the axial flow at avelocity vsz and the absorption in the Larmor peak increase monotonically(Kopu, 2006). Thus the maximum Larmor peak height htw in Figure 14 is


0 50 100 150 200 250Time (s)

NM

R a

bsor

ptio

n (m

V)

exp(2t/tLar)

htw

10%

80%

10%

0.8tCF

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.2

1.3Ω

(ra

d/s)

ΩcAB

Larmor peakcf peakT = 0.50TcP = 29 bar

FIGURE 14 NMR signal responses of the propagating vortex front and the twistedvortex state behind it. Here the turbulent burst is started with the Kelvin-Helmholtzinstability of the AB interface. Top panel: it is triggered by increasing � by a smallamount �� across the critical value �cAB at 1.3 rad/s. Main panel: two absorptionresponses are shown, which have been recorded with the bottom detector coil atconstant but different values of magnetic field. The responses are from twoconsecutive identical measurements to allow direct comparison of signal amplitudes.The counterflow peak height (thick grey line) shows the KH trigger �� and a rapidcollapse when the front moves through the coil. The time interval from t = 0 (when�(t) = �cAB) to the start of the collapse measures the flight time of the front fromthe AB interface to the closer end of the detector. The moment when the peak heightreaches zero corresponds to the point when the front has passed through the rearend of the coil. The time required for the collapse, ∼τCF, measures the width of thevortex front. The second signal (thin black line) is recorded close to the Larmor edgeand is sensitive to the longitudinal velocity vsz which is generated by the twistedvortex state (Figure 21). Its sudden steep rise at t ≈ 30 s is caused by the passage ofthe first helical sections of the twisted state through the coil. Its later exponentialdecay reflects the unwinding of the twist, which starts when the front has reachedthe end plate of the cylinder and the vortex ends begin to slip along the flatsurface.

reached when vsz reaches its largest value inside the detector coil. This hap-pens when the most tightly spiralled section of the twisted cluster (which isjust behind the front, as seen in Figure 21) passes through the middle of thedetector coil. When the front arrives at the bottom end plate of the cylin-der, the twist starts to relax since the vortices have to obey the boundarycondition on the flat end plate, where they slip to reduce their length andwinding. The unwinding produces the exponentially relaxing absorption


with time constant τLar. The signal decay continues down to the absorp-tion level characteristic of the equilibrium vortex state, with an array ofrectilinear vortex lines. At their other end, the vortices, after crossing theAB interface, have a continuation as doubly quantised A-phase vortices(Hänninen et al., 2003).

To interpret measurements on the front velocity, it is of importance thatthe structure of the twisted state and the front itself are known. Informationon these characteristics can be obtained by measuring quantitatively thevarious features denoted in Figure 14 as a function of temperature.

3.3 Velocity of vortex front

A striking consequence from the twisted state is the appearance of super-flow directed along the helically spiralling vortex cores. This situation isreminiscent of a “force-free” vortex configuration, where all the flow isdirected along the vortex core. Such a structure is expected to be stableup to some instability limit, similar to the Glaberson limit of a rectilin-ear vortex array in parallel flow (Section 2.4). In the twisted state, whichis uniform in the axial and azimuthal directions, the superflow has bothan azimuthal component at the velocity vsφ and an axial component atvsz, which depend on the radial coordinate r and are described by theexpressions (Eltsov et al., 2006b):

vsφ(r) = (� + Qv0)r1 + Q2r2 , vsz(r) = v0 − Q�r2

1 + Q2r2 . (9)

Since the net flow through the cross-section of the cylindrical con-tainer should vanish, from this condition for vsz one finds that v0 =(�/Q)[Q2R2/ ln(1 + Q2R2) − 1]. The axial flow is directed along the vortexexpansion direction close to the cylindrical wall and in the opposite direc-tion closer to the centre. In practice, there has to exist also a radial velocityvsr since any laboratory example of the twisted state is nonuniform. In thecase of a propagating vortex front, the wave vector Q has its maximumvalue close to the rear end of the front and decreases to zero at the bottomand top end plates of the sample. As seen in Figure 14, the twisted stateprominently changes the line shape of the NMR spectrum and it is theaxial superflow which here has the strongest influence.

As discussed in Section 2.2, in vortex-free counterflow, the end point ofa single vortex moves on the cylindrical sidewall roughly with the longitu-dinal velocity vLz ≈ α�R while its azimuthal velocity is vLφ ≈ −(1 − α′)�R(in the rotating frame). Thus the wave vector of the spiral trajectory isQ = |vLφ/(RvLz)| ≈ (1 − α′)/(Rα) = ζ/R. If this value is used as an estimate


for the wave vector of the twisted state, it follows that the helical windingof the vortex bundle becomes tighter with decreasing temperature. Thetighter twist increases the flow velocities in Equation (9), that is, flow par-allel to the vortex cores is enhanced, which will ultimately destabilise the‘force-free’ twisted-cluster configuration. Nevertheless, twisted-clusterpropagation appears to persist even below 0.2 Tc. The twist is removedby the slip of the vortex ends along the flat end plates of the cylinder,which generates the exponentially relaxing absorption in Figure 14 withthe time constant τLar.

In Figure 15, left panel, the measured temperature dependence of themagnitude of the twist is plotted in terms of the maximum height of theLarmor peak htw, normalised to the height of the Larmor peak at � = 0at the same temperature (left vertical axis). Two experimental setups withslightly different specifications were used in these measurements. The lineshape of the NMR absorption spectrum in the Larmor region depends bothon the magnitude and homogeneity of the magnetic polarisation field. Inthe two setups, the homogeneities varied by a factor of two, which isbelieved to explain the differences in the absolute values between the two

0.3 0.4 0.5 0.6 0.7 0.8

T/Tc

0

1

2

3

4

5

h tw

/hL(

Ω =

0)

Setup 1Setup 2Simul.

0

0.5

1

1.5

QR

T/Tc

0.30.40.3 0.5 0.60

1

2

3

4

5

Vfτ

CF, c

m

FIGURE 15 Left: magnitude of the twist as a function of temperature. Themeasurements are performed using the bottom spectrometer. Setups 1 and 2 refer tomeasurements with the detector coils positioned in two different sets of positionsalong the sample cylinder; in Figure 13 setup 2 is depicted. (◦,�): the measured ratioof the maximum amplitude htw of the Larmor peak in the twisted state to theamplitude hL(� = 0) of the Larmor peak in the nonrotating sample is plotted on theleft axis. (�): the maximum value of the twist wave vector Q, obtained fromsimulation calculations, is plotted on the right axis. The solid curve shows the fitQR = 0.7(1 − α′)/α. The dashed curve shows the minimum Q at which the vortexfront still propagates in a thin steady-state configuration. Right: apparent thicknessof the vortex front τCFVf as a function of temperature, as determined frommeasurements triggered with the Kelvin-Helmholtz instability (Figure 13). Thesolid line is the prediction of the model in Equation (11).


data sets (for details, we refer to Eltsov et al. (2008)). Nevertheless, it is seenhere that the twist increases towards low temperatures as expected, butonly until a maximum at 0.45Tc, whereas below 0.45 Tc it abruptly startsto decrease.

The nonmonotonic temperature dependence of the twist is confirmedin numerical calculations: the value of the twist wave vector behind thefront, as determined from a fit of the calculated velocity profiles to Equa-tion (9) (Eltsov et al., 2008) and plotted in Figure 15 (left panel, right verticalaxis), also peaks at 0.45Tc. Two reasons can be suggested for the change intemperature dependence at 0.45 Tc. First, the twist can relax via reconnec-tions between vortices in the bundle, which become more frequent withdecreasing temperature below 0.4 Tc (cf. Figure 25). Second, the source ofthe twist is at the vortex front, while the sink is at the end plate of the cylin-der where the twist vanishes because of the boundary conditions. Fromthere the relaxation of the twist advances in a diffusive manner along thetwisted bundle. The effective diffusion coefficient increases as the temper-ature decreases (Eltsov et al., 2006b), and thus the faster diffusion limitsthe maximum twist in a finite-size sample at low temperatures. However,overall the stability of twisted-cluster propagation appears to be a compli-cated question at temperatures below 0.3 Tc, where it controls the averagenumber of vortices threading through each cross section of the columnbehind the front.

The properties of the twist in Figure 15 roughly agree with the estimatethat the front velocity can be approximated with the longitudinal velocityof a single vortex expanding in vortex-free rotation, vLz ≈ α�R. However,this simplification suffers from the following difficulty: ahead of the front,the vortex-free superfluid component is at rest and the effective counter-flow velocity might really be approximated with v = �r, but behind thefront, the density of vortices is close to equilibrium and vsφ ≈ vnφ. In thiscase, a vortex, which has fallen behind in the motion, feels a much reducedcounterflow and continues to fall more behind. Therefore the thickness ofthe front should increase with time. The explanation to this dilemma is thatbehind the front the superflow induced by the twisted vortex bundle hasto be taken into account. The longitudinal expansion velocity should nowbe modified to vLz = α [vnφ(R) − vsφ(R)] + (1 − α′)vsz(R). Since here vsz(R)

is oriented in the direction of the front propagation and vsφ(R) < vnφ(R)

in the twisted state, the longitudinal expansion velocity Vt of the vorticesin the tail of the front is enhanced. This velocity can be estimated takingvsz(R) and vsφ(R) from Equation (9):

Vt = α�R[

1 + 1 − α′

α

1Q R

] [1 − Q2R2

(1 + Q2R2) log(1 + Q2R2)

]. (10)


Equation (10) has a maximum as a function of the Q vector. If (1 − α′)/α <

1.9 (which corresponds to T > 0.46 Tc according to the measurements ofBevan et al. (1997)), the maximum value of Vt is less than the velocity of theforemost vortices Vf ≈ α�R. In these conditions, the thickness of the frontincreases while it propagates. When T < 0.46 Tc, a wide range of Q valuesexists for which formally Vt � Vf. The minimum possible value of Q isshown in Figure 15 (left panel) as the dashed curve. In these conditions,the front propagates in a steady state ‘thin’ configuration.

Experimentally, the decay time of the counterflow peak τCF in Figure 14can be used to extract the front thickness. The decay starts when the headof the front arrives at the closer edge of the detector coil and it is over whenthat part of the front leaves the far edge of the detector coil where the coun-terflow is not sufficient to generate a nonzero absorption response. Theproduct τCFVf has the dimension of length and can be called the appar-ent thickness of the front. At higher temperatures, the actual thicknessof the front grows with time. Here the apparent thickness depends onthe distance of the observation point from the site of the turbulent burstand on the rate at which the thickness increases and vortices fall behind.With decreasing temperature, the front starts to propagate as a thinsteady-state structure and ultimately its apparent thickness decreases toequal the height of the pick-up coil (hc = 9 mm, Figure 7) and remainsthereafter approximately constant.

Measurements of the apparent thickness of the front are presented inthe right panel of Figure 15. At T > 0.45Tc, τCFVf > hc and the apparentthickness increases with increasing temperature. At 0.45 Tc, the appar-ent front thickness becomes comparable with the height of the detectorcoil and thereafter at lower temperatures remains at that value. Assum-ing that initially at the turbulent burst the front is infinitely thin, we canwrite

τCF = hb + hc

V∗t

− hb

Vf, (11)

where hb is the distance from the site of the turbulent burst to the nearestedge of the pick-up coil, and V∗

t is the expansion velocity at the positionin the front where the NMR signal from the counterflow vanishes. Giventhat the latter condition roughly corresponds to vsφ ∼ (1/2)vnφ, we takeV∗

t = (Vt + Vf)/2 if Vt < Vf and simply V∗t = Vf otherwise. Using Vt from

Equation (10) and the simple estimates QR = (1 − α′)/α and Vf = α�R,we get from Equation (11) the solid line in Figure 15 (right panel), whichis in reasonable agreement with experiment.

The rapid change in the counterflow peak height during the passageof the vortex front through the detector coil provides a convenient sig-nal for measuring the propagation velocity Vf . Examples of these signals


0

1N

orm

alis

ed N

MR

abs

orpt

ion

0 100 300Time (s)

0

1

0

1T � 0.40Tc

T � 0.29Tc

a � 0.23

a � 0.041

T � 0.22Tca � 4.8·10�3

200

FIGURE 16 Measurement of vortex front propagation. The NMR absorption in thecounterflow peak is monitored as a function of time, after triggering theKelvin-Helmholtz instability at t = 0 (cf. Figure 14). The instability is started inconstant conditions in the vortex-free state at 1.2 rad/s by increasing � in one smallstep above the critical value �cAB, when a magnetic field stabilised A-phase layer ispresent (topmost panel), or by increasing the magnetic field stepwise above HAB atconstant � (two lower panels). The two signal traces denote the top (thin noisy line)and the bottom (thick line) detector. Since the flight paths for the upper and lowersample sections are almost equal, the two traces display almost identical flighttimes.

are shown in Figure 16. They have been measured at different temper-atures to illustrate how the temperature dependence of Vf is expressedin the practical measurement. All three examples have been measuredusing the externally triggered KH instability to start the turbulent burst.In Figure 16, the vortex fronts have traveled a distance of ∼4 cm, beforethey pass through the detector coil, and thus have already acquired theirsteady-state thin-front configuration.

Measurements on the front velocity Vf are shown in Figure 17.As a function of temperature, two different regimes of front propa-gation can be distinguished, the laminar and turbulent regimes. Thecrossover between them is gradual and smooth. This is in sharpcontrast to the sudden onset of bulk turbulence (as a function oftemperature around ζ ∼ 1) in injection measurements in the same cir-cular column, when a bundle of closely spaced seed vortex loopsescapes across the AB interface in a Kelvin-Helmholts instability event(Figure 4). A sharp transition with a clearly defined critical velocityis the usual case, for instance in all measurements with mechanicalvibrating objects (in the regime ζ > 1) as a function of drive and flowvelocity (Vinen and Skrbek, 2008). The smooth crossover here in front


0.2 0.3 0.4 0.5 0.60.0

0.2

0.4

0.6

0.8

ABOrifice

�(T )

3 4 5 6 7Tc/T T/Tc

1023

1022

1021

Vf/(

ΩR

)

Vf/(

ΩR

)

�(Bevan et al.)

Turbulent

Quantum

Laminar

Quasi-classical

Twist

Simul

FIGURE 17 Normalised velocity Vf/�R of vortex front propagation in a rotatingcolumn. All externally controlled variables are kept constant during this measurementwhere the initial state is vortex-free rotation and the final state an equilibrium arrayof rectilinear vortices. Main panel: the assignments of different hydrodynamic regimesrefer to the dynamics in the front motion. The open circles denote measurements inwhich the turbulent burst is started from the AB interfaces in the middle of thecolumn. The squares refer to the case where the front is started at the orifice andthen moves upward through the entire column. The large filled diamonds mark resultsfrom numerical calculations. The dashed line represents the mutual frictiondissipation α(T ) measured by Bevan et al. (1997) and extrapolated below 0.35 Tc withexp(−�/T ) (Todoshchenko et al., 2002). The dash-dotted curve takes into account thetwisted vortex state behind the propagating front (Equation (12)). The solid curvedisplays the theoretical model from Section 3.5 which includes corrections from thetwisted vortex state, turbulent energy transfer and quantum bottleneck. Left panel:this semilog plot shows that the analytically calculated model provides a reasonablefit to the low temperature data.

propagation may be a special property of the circular column where vor-tex polarization along the rotation axis is always � 90 %. Nevertheless,smooth crossovers have been observed before, for instance from linearto turbulent wave acoustics in second sound propagation in a circularcylinder as a function of driving amplitude (Kolmakov et al., 2006).

Above 0.4Tc in the laminar regime, the results in Figure 17 are consistentwith the earlier measurements of Finne et al. (2004c). Here the single-vortex dynamics apply when inter-vortex interactions can be neglectedand Vf ≈ α�R (Section 2.2). At closer inspection, it is noticed that thedata for the normalised front velocity vf = Vf/�R lie on average belowα(T) (dashed curve). The reason is that behind the front the vortices arein the twisted state and not as rectilinear vortex lines in an equilibriumarray. If we integrate the kinetic energy stored in the twist-induced flow


(Equation (9)) over the sample cross-section, the result is found to be

vf,lam = 2[1/ log(1 + ζ2) − ζ2]α . (12)

After including this reduction from the twisted state, the agreement isimproved (dash-dotted curve). In the turbulent regime below 0.4Tc, thedata deviate with decreasing temperature more and more above theextrapolations from the laminar regime. Eventually at the lowest tem-peratures, the measurements become temperature independent, with apeculiar transition from one plateau to another at around 0.25 Tc. Thesefeatures are attributed to turbulent dynamics and are analysed in the nextsections in more detail.

The measured properties of the propagating vortex front are confirmedqualitatively in numerical calculations. They show that in the laminarregime the thickness of the front grows with time, but with decreasing tem-perature the twist increases (Figure 15) and finally at about 0.45 Tc the thinsteady-state front configuration is established, with a time-independentthickness roughly equal to the radius of the sample. Below 0.45 Tc, thetwist decreases with decreasing temperature but remains within the limitswhere the twist-induced superflow is sufficient to maintain a thin time-invariant front configuration. The calculated front velocity in Figure 17approximately agrees with the measurements down to about 0.22 Tc. It istherefore instructive to analyse the calculations to identify where and bywhat mechanisms turbulent losses occur in the rotating column. This willbe discussed in the next section.

3.4 Numerical Calculation of Turbulence in VortexFront Propagation

The promising agreement of the calculated vortex front velocity vf withthe measurements in Figure 17 suggests that numerical calculations couldprovide useful guidance for the interpretation of the measurements inSection 3.3 and for constructing the analytic theory in Section 3.5. Our mainquestion is the following: why are the numerical results on vf deviatingwith decreasing temperature more and more above the mutual-friction-controlled extrapolation from the laminar regime? For simplicity, we splitthe discussion of vortex motions to three different length scales: (i) largescales of order ∼R, where turbulent fluctuations become visible as vari-ations in the number and distribution of vortices, (ii) the inter-vortexscale ∼�, where the presence of Kelvin-waves on individual vortex linescan be seen to grow and (iii) small scales, where reconnections betweenneighbouring vortices might occur and excite turbulent fluctuations on


individual vortex lines. In contrast to the more usual studies of turbu-lent tangles, which generally monitor their free decay as a function oftime when the external pumping is switched off, we are here dealingwith highly polarised steady-state motion along the rotating column. Thismotion appears to remain intact down to the lowest temperatures butbecomes dressed with turbulent fluctuations in growing amounts towardsdecreasing temperatures.

Two examples are shown in Figure 18, which illustrate the developmentwith decreasing temperature. These configurations at 0.4 Tc and 0.3 Tc inan ideal column of radius R = 1.5 mm rotating at � = 1 rad/s have beencalculated using the techniques described in Section 2.5. Similar results at

t 5 140sT 5 0.3Tc

t 5 80sT 5 0.4Tc

28 m

m

FIGURE 18 Calculated vortex propagation at 0.3 Tc and 0.4 Tc. The motion is startedfrom the bottom end of the cylinder by placing the equilibrium number of vortices inthe form of quarter loops between the bottom end plate and the cylindrical sidewall.On the right at 0.4 Tc, the front has traveled for 80 s to a height z ≈ 28mm above thebottom end plate in a cylinder of 3mm diameter rotating at 1 rad/s. The zoom on thefar right shows the vortices in the front and immediately below in more detail. On theleft at 0.3 Tc, the same distance is covered in 140 s. Here the vortices appear morewrinkled, owing to short-wavelength Kelvin-wave excitations. The vortices are tightlytwisted below the front but become straighter and smoother on approaching thebottom end plate. Many vortices can be seen to connect to the cylindrical sidewallalso below the front. Their average length is shorter than the distance from the frontto the bottom end plate, although the number of vortices N(z) threading througheach cross-section of the cylinder below the front is roughly constant and comparableto that in the equilibrium vortex state: N(z) � Neq. The average polarisation along thevertical axis is high, ∼90%.


0.4 Tc and higher have been reported in the review by Finne et al. (2006b).The calculations are started from an initial configuration with roughlythe equilibrium number of vortices placed as quarter loops between thebottom end plate and the sidewall of the cylinder. During the subsequentevolution, the propagating vortex front is formed and the twisted clusterstarts to acquire its shape (Hänninen, 2006).

Above 0.45 Tc, the motion is laminar with relatively smooth vortices,which only twist at large length scales. Below 0.45 Tc, the thickness ofthe front (in the axial direction) settles at �(r) � r d, where the parameterd ∼ 1. The large-scale characteristics as a function of temperature from 0.3to 0.6 Tc are displayed in Figure 19 for the setup of Figure 18. The counter-flow energy E(z) is obtained by integrating the momentary distributionof counterflow v(r,φ, z, t) over each cross section z of the column, whilethe polarization pz(z) is similarly derived by integrating over the vorticesthreading through this cross section. With decreasing temperature, thefront acquires more and more turbulent features which become visibleas increasing small-scale structure: Kelvin-waves, kinks and inter-vortex

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

0.6 Tc

0.5 Tc

0.3 Tc

0.4 Tc

vortex-free flow

vortexfront

twistedvortex cluster

R 5 1.5 mmL 5 40 mmΩ 5 1 rad/s

z (m

m)

0

5

10

15

20

25

30

35

40

0.2 0.4 0.6 0.8 1

0.6 Tc

0.5 Tc

0.4 Tc

0.3 Tc

z(m

m)

Polarization pz2 (z)

2pz

25∫ dj / ∫ dj∧⎛ ⎞⋅⎝ ⎠s z

∧

Counterflow energy (z)

(z)5∫ � 2nsd 2r / ∫ (Vr)2d 2r

FIGURE 19 Axial distributions of counterflow energy E (z) (see Eq. (8)) andpolarization pz(z) (both in normalized units), calculated for the vortex propagation inFigure 18. On the left E (z) is seen to drop steeply within the narrow vortex front atthe temperatures 0.4 and 0.3 Tc. Here the front is in steady-state time-invariant motionand its velocity provides a direct measure of dissipation. Above 0.45 Tc some vorticestend to fall more and more behind during the motion and the shape of the frontbecomes more extended with time. As seen in Figure 18, small-scale structure fromKelvin waves accumulates increasingly on the vortices below 0.45 Tc, but as shown inthe panel on the right, the polarization pz(z) remains always high behind the front.


reconnections. These small-scale fluctuations exist on top of a stronglypolarised vortex orientation, which is preserved even at 0.2 Tc. The highpolarisation along the rotation axis and the organised configuration of thetwisted cluster is expected to reduce loop formation and self-reconnectionon individual vortex lines, but it also suppresses reconnections betweenneighbouring vortices from what one would find in turbulent tangles. Theamount of twist, in turn, is reduced by the unwinding from the slip of thevortices along the flat bottom end plate of the column.

The comparison of the two examples in Figure 18 is continued inTable 1. As a rule, we expect that with decreasing mutual-friction-dampingturbulent disturbances are expected to cascade down to ever smaller lengthscales. In Table 1, this is examined by defining the quantity δ which mea-sures on an average the distance over which the vortex front moves duringthe time interval between two inter-vortex reconnections. Since the axialmotion of the front slows down and the reconnection rate increases withdecreasing temperature, δ decreases rapidly below 0.4 Tc. Thus, δ providesa measure between mutual friction and reconnections. The bottom linein Table 1 uses the dimensionless ratio δ/� to characterise the relativeimportance of mutual friction dissipation to reconnection losses. Whenδ/� > 1, the energy loss from reconnections, which presumably exciteKelvin-waves, is not important in the total energy balance of the prop-agation. At 0.4 Tc, where δ/� ∼ 2, this is what one expects. If δ/� < 1, as isthe case below 0.3 Tc, then it becomes possible that inter-vortex reconnec-tions might play a role in the total energy balance. Table 1 thus hints that

TABLE 1 Comparison of vortex front propagation at 0.3 Tc and 0.4 Tc. The bottomline of the table shows that below 0.4 Tc front propagation is rapidly moving into thequantum regime. The calculations are for a cylinder of radius R = 1.5mm and lengthh = 40mm, which is filled with 3He-B at a liquid pressure of P = 29 bar and rotatesat an angular velocity � = 1 rad/s. The number of vortex lines is the average througheach cross-section of the cylinder over the length of the twisted cluster. Themaximum resolution of the calculations is 0.05mm.

T/Tc 0.3 0.4

Mutual friction parameter α 0.040 0.18Mutual friction parameter α′ 0.030 0.16Front velocity Vf 0.12�R 0.22�RFront velocity Vf (mm/s) 0.18 0.33Total Reconnection rate (event/s) 300 130Number of vortex lines/cross-section 125 150Reconnection rate per line 1/τ [events/(line s)] 2.4 0.87Front shift δ during τ (mm) 0.075 0.38Interline separation � (mm) 0.20 0.19Ratio δ/� 0.36 2.0


new phenomena are expected to emerge on length scales approaching theinter-vortex distance �.

The characteristics of the propagating vortex front and the trailingtwisted cluster are further illustrated in Figures 20–22. In Figure 20 it is themodulus of the total velocity of the superfluid component, |vs(r,φ, z, t) |,whose [r, z] profiles are displayed. The top panel shows the velocity 〈vs〉t,φin the laboratory coordinate frame, averaged over time t and azimuthalangle φ, while the bottom panel displays its mean fluctuation amplitude[〈(vs − 〈vs〉t,φ)

2〉t,φ] 12 . In Figure 21, it is the axial component vsz and in

Figure 22 it is the radial component vsr, which are analysed in the samefashion. The largest component vsφ is omitted since its profiles are so sim-ilar to those of |vs |. In fact, at large radii r � R/5, the profiles of the twovelocities as a function of [r, z] are almost indistinguishable, while at smallradii r � R/5 the axial velocity vsz in Figure 21 is the main contributor to|vs |.

The vortex front becomes clearly defined in Figures 20–22. For instance,in Figure 20, the steep almost linear rise in |vs | in the interval −1 < z/R < 0signifies the transition from the nonrotating state vsφ = 0 at z > 0 to almostequilibrium rotation at vsφ ≈ �r at z < −R. This is the vortex front with itsnarrow thickness ∼R and strong shear flow, created by the vortices termi-nating within the front on the cylindrical sidewall perpendicular to the axisof rotation. Below the front, the number of vortices threading through anycross-section of the cylinder remains roughly constant. This is seen fromthe fact that at 0.3 Tc the maximum value of vs, 〈vsφ〉max ≈ 0.6�R, remainsstable over the entire length of the twisted cluster, starting from z/R < −1immediately behind the front. At higher temperatures some vortices tendto fall behind the front and thus at 0.4 Tc one finds that vs slowly increasesto 〈vsφ〉max ≈ 0.75�R at z/R < −5. Note also that radially the azimuthalflow vsφ increases monotonously behind the front up to the edge of thecluster at ∼0.9 R and then slightly decreases towards the cylindrical wall.

The fluctuations of |vs | around its mean value, as shown in the bottompanel of Figure 20, are substantial of order 30 % in the sharp peak cre-ated by the vortex front. The same applies to other velocity components;their fluctuations also tend to be largest in the front region −1 < z/R < 0and typically only half as large over the length of the twisted cluster.In Figures 20–22, the fluctuations are sampled at a relatively slow rateof 2 Hz. As seen in Figure 18, in the front, the vortices are not perfectlydistributed: it is this disorder in the structure of the front, and the vari-ations in its number of vortices which gives rise to the fluctuation peak.In Figure 23, the radial distribution of the fluctuations in |vs | in the frontregion −1 < z/R < 0 is analysed. As expected, they grow rapidly towardslarge radii. In Section 3.5, we make use of this radial distribution.

However, not only large-scale disorder contributes to velocity fluctu-ations but also Kelvin-waves start to expand on individual vortices at


021222324

r 5 R

2526

0212223242526

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

z/R

0

0.87 R 0.93 R4 R/5

3 R/5

2 R/5

R/5

0

0.05

0.1

0.15

0.2

z / R

r 5 R

r 5 0

0.93 R

4 R/53 R/5

2 R/5R/5

h[�2

h�it,φ

]2 i t,φ

,/ΩR

!ww

ww

ww

wh�i

t,φ,/Ω

R

FIGURE 20 Axial and radial distributions of the velocity |vs(r,φ, z, t) | and itsfluctuations in vortex front propagation. Both quantities are normalised with respectto �R and are expressed in the laboratory frame in terms of their averages, 〈vs〉t,φ (top

panel) and [〈(vs − 〈vs〉t,φ)2〉t,φ] 12 (bottom panel), by integrating over the azimuthalcoordinate φ and time t. The different contours are plotted as a function of z at fixedradial value, r = 0, R/5, 2 R/5, . . .. They are calculated, for example, at 0.3 Tc inFigure 18, over the time interval from 60 to 80 s after starting the front motion. Att = 80 s, the front has climbed to a height z(80 s) ≈ 16mm (here placed at z/R = 0).Both plots are generated from vortex configurations, which are saved every 0.5 s. Thesteep change in the top panel in the interval −1 < z/R < 0 signifies the front with itsnarrow thickness �z ∼ R. Similarly in the bottom panel the amplitude of fluctuations(sampled at 2 Hz) reaches a sharp maximum in the front region −1 < z/R < 0 at largeradii r � 3 R/5. This peak is twice larger than the flat values from behind the front inthe region of the twisted cluster. At 0.4 Tc (the case of Figure 18, right), the fluctuationpeak from the front is 14% lower.


0�1�2�3�4�5�60.2

�0.20.14

0.12

0.1

0.08

0.06

0.04

0.02

0

0.1

�0.1

0

0�1�2�3�4�5�6

z/R

z/R

r � R

r � R

r � 0

r � 0

3 R/5

3 R/5

2R/5

2R/5

4R/5

4R/5

R/5

R/5

([� z

�h� z

i t,φ]2 )

t,φ,/Ω

R��

��

��

��

�h� z

i t,φ,/Ω

R

FIGURE 21 Axial and radial distributions of the axial velocity component vsz(r,φ, z, t)and its fluctuations. The profiles have been generated in the same calculations at0.3 Tc as Figure 20. This component is generated by the helically wound sections in thetwisted cluster. The slowly unwinding twist at the bottom end plate (on the left ofthe figure) causes the characteristic linear increase in |vsz | towards the right wherenew twist is continuously formed by the spirally winding motion of the vortex front.Note that vsz changes sign since it is directed antiparallel to the propagation directionof the front in the central parts r � 2 R/3 and parallel at larger radii. Its maximummagnitude is in the centre just behind the front where it is the dominant componentin |vs |. Its largest fluctuations are at large radii within the front region.

temperatures � 0.3 Tc, as seen in Figure 18 (Hänninen, 2006). In Figure 24,a momentary vortex configuration (when the front has reached the heightz ≈ 28 mm in Figure 18) is broken down in curvature radii Rc and theaverage curvature 〈R−1

c 〉 is plotted as a function of z. At T ≥ 0.4 Tc, thesharp peak at the front is caused by the vortices curving to the cylindricalsidewall with Rc ≈ R. However, at 0.3 Tc, the Kelvin-wave contributionat shorter length scales becomes dominant and the average radius of cur-vature drops to 〈Rc〉 ≈ 0.4 mm. The characteristic Kelvin-wave frequencyωk ∼ κk2 corresponds at this length scale to 0.4 Hz (which is less thanthe sampling frequency of 2 Hz). Thus the dominant Kelvin-waves areincluded in the velocity fluctuations in Figures 20–22. Nevertheless, in


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.01

0

�0.01

�0.02

�0.03

�0.04

0�1�2�3�4�5�6

0�1�2�3�4�5�6

z/R

z/R

Rr � 0

r � 0

r �R

0.93 R

0.87 R

0.93 R

4 R/5

4 R/5

3 R/5

3 R/5

2 R/5

2 R/5

R/5

R/5

h� ri t,φ

/ΩR

([� r

�h� r

i t,φ]2 )

t,φ,/Ω

R��

��

��

��

FIGURE 22 Axial and radial distributions of the radial velocity componentvsr (r,φ, z, t) and its fluctuations in vortex front propagation at 0.3 Tc in Figure 18. Thiscomponent provides the return currents to the axial component vsz . It is an order ofmagnitude smaller than either the azimuthal vsφ or the axial vsz component. It isdirected inward towards the central axis in the narrow front region, while the outwardflowing radial return currents are distributed more evenly along the length of thetwisted cluster. The fluctuations in vsr are largest close to the cylinder axis andsteadily decrease as a function of r. As a function of z, the maximum fluctuations arejust behind the front where vsr changes sign.

these figures, Kelvin-waves represent only a small part of the total fluctu-ations 〈(vs − 〈vs〉t,φ)

2〉t,φ in the sharp peak created by the front. This is seenfrom the fact that the peak is localised in the z direction within the vor-tex front at −1 < z/R < 0, while the Kelvin-wave excitations in Figure 24extend half way down the twisted cluster to z/R ∼ −10, as can also bedirectly seen from Figure 18.

Figure 25 shows that also reconnections between neighbouring vor-tices become rapidly more frequent at temperatures � 0.3 Tc (see alsoHänninen, (2006)). The reconnections peak in the region around z/R �−1, where the front ends and the twisted cluster starts. The increasing


0 0.2 0.4 0.6

(r/R)2

0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

T � 0.3 Tc

T � 0.4Tc

<(��

h�it,φ

)2> t

,φ/(

ΩR

)2

FIGURE 23 Radial distribution of velocity fluctuations within the propagating vortexfront from Figure 20. The normalised square of the velocity deviation from theaverage is plotted on the vertical scale, 〈(vs − 〈vs〉t,φ)2〉t,φ/(�R)2, versus the square ofthe normalised radial position, (r/R)2. The averages in the angled brackets, besideshaving been integrated over time t and azimuthal coordinate φ (or front motion),represent an average over the interval −1 < z/R < 0. The fluctuations are seen toincrease rapidly towards large radii, up to where the vortex-free annulus starts atRv ≈ 0.87 R.

reconnection rate with decreasing temperature is one factor which helpsto reduce the helical twist behind the front and might be one reason why themeasured twist appears to be reduced towards low temperatures below0.45 Tc (Figure 15).

A further characteristic of front propagation in the rotating columnis that fluctuations in vortex length are small since these are mainlyrestricted to the transverse plane. The total vortex length L(t) in themoving front and in the twisted cluster behind it increases linearly intime and thus the fluctuations can be expressed with respect to a lin-early increasing fitted average Lfit(t). The average deviation from themean 〈�L〉 = [〈(L − Lfit)

2〉] 12 turns out to be small. In the conditions of

Figure 18 at 0.3 Tc, one obtains �L/L ∼ 10−3. This is as expected sincefluctuations in length are energetically expensive in polarised vortexmotion.

Finally, we point out two features which are seen in the calculations,which both grow in prominence with decreasing temperature below 0.3 Tcbut which have not yet been searched for in measurements. The calcula-tions are started with the number of seed vortices equal to or larger than


0

0.5

1

1.5

2

2.5

⟨c�

1 ⟩R

3

3.5

z (mm)0

0.3Tc

0.4Tc

0.5Tc

0.6Tc

�5�10�15�20�25

P � 29 bar

FIGURE 24 Distribution of curvature radii along vortices during front propagation,calculated for the setup in Figure 18. The curvature is defined as the inverse of thelocal vortical curvature radius Rc, that is R−1

c = |d2s(ξ, t)/dξ2|, where ξ is thecoordinate along the vortex core. The curvature is calculated on an equidistant lineargrid along each line vortex. On the vertical axis, the average curvature 〈R−1

c 〉 isplotted as a dimensionless quantity 〈Rc

−1〉 R as a function of z at differenttemperatures. The averages are taken over the cross-section of the cylinder and overan interval �z = 2mm in length centred at z. The sharp peak at the very front(z ≈ −2mm) at temperatures T ≥ 0.4 Tc represents the vortices curving to thecylindrical sidewall with Rc ≈ R. In contrast, at 0.3 Tc, small-scale curvaturedominates, the average radius of curvature has dropped to Rc ≈ 0.4mm and extendsfrom the front well inside the twisted cluster.

Neq, but nevertheless, the number of vortex lines does not remain stableat Neq during steady-state propagation.1

The first feature concerns the number of vortices which in freesteady-state propagation thread through any cross-section of the col-umn behind the front. This number is roughly constant but less than inequilibrium (for instance in Figure 18 at 0.3 Tc, there are about 130 vor-tices per cross-section while Neq ≈ 160 vortices). In the simplest modelof the twist in Equation (9), the number of vortices in the twistedstate is smaller than in the equilibrium vortex state by the factor ≈(QR)2/

([1 + (QR)2] ln [1 + (QR)2]). This reduction is of correct order ofmagnitude compared to that seen in the calculations at � 0.3 Tc. In additionto improving the stability of the twisted state, the reduction in the numberof twisted vortices decreases the axial flow velocity vsz (Figure 21), whichhas the effect of reducing the longitudinal front velocity vf and counteracts

1 In the equilibrium vortex state Neq ≈ π(R − deq)2 nv, where the width of the vortex-free annulus around

the central cluster is deq ≈ n−1/2v (Ruutu et al., 1998b).


0 20 40 60 80 100 120

0.6Tc

0.5Tc

0.4Tc

0.3Tc

T/Tc

140

Time (s)

Cum

ulat

ive

num

ber

of a

ll re

conn

ectio

ns�

104

Front at top plate

0.25 0.35 0.45 0.55 0.65

0

100

200

300

400

Rec

onne

ctio

n ra

te (

1/s)

0

1

2

3

4

FIGURE 25 Main panel: cumulative number of all vortex reconnection eventsduring vortex front propagation in the setup of Figure 18, as a function of time. Thereconnections occur between two different vortices and mainly in the twisted clusterwhere they reach a maximum close behind the front. This process does not changethe number of vortices. The vertical arrows indicate the moment when the frontreaches the top end plate (at height h = 40mm). The reconnections are hereseen to increase rapidly at low temperatures, reaching a rate of 300 s−1 or ∼1reconnection/(vortex s) at 0.3 Tc. Inset: the roughly constant reconnection rate fromthe main panel (when the front is in steady-state propagation before the end plate isreached) plotted as a function of temperature.

its increase above the laminar extrapolation in Figure 17. While the numberof vortices per cross section of the twisted cluster decreases with decreas-ing temperature below 0.3 Tc, the density of vortices, nevertheless, remainsconstant over the cross section (but less than the equilibrium value:nv � nv,eq = 2�/κ). When the front finally reaches the upper end plate inFigure 18 and the twist starts to relax, simultaneously N gradually recoversand approaches Neq from below.

The second feature of the calculations is that in steady state propagationthe average length of the vortices is less than that of the twisted cluster(which means that the total number of individual vortices in the columnmay be well above Neq). For instance at 0.3 Tc in Figure 18, the averagelength is ∼15 mm and only ∼6 mm for those vortices with both ends on thecylindrical sidewall. Nevertheless, the polarisation of all vortices along zis high, ∼90 %. After the front has reached the upper end plate in Figure 18and the twist starts relaxing, the short vortices provide an ample storagefrom which to add more vortices so that Neq is reached.

To summarise the calculations, we return to our starting point, namelythe increasing deviation of the calculated front velocity above the laminarextrapolation below 0.4 Tc (Figure 17). The analysis of the calculated vortex


configurations shows increasing turbulent disorder at large length scale∼R, growing Kelvin-wave amplitudes on individual vortices on scales ∼�

and the presence of inter-vortex reconnections. Combined these changesfrom the increasing turbulent influence with decreasing temperature makeup for the difference, in the presence of a still finite mutual friction dis-sipation. An additional sink of energy is the cascading of Kelvin-waveexcitations below the resolution limit of the numerical calculations (usu-ally ∼0.05 mm), which are lost from the energy balance. The calculationsare in reasonable agreement with measurements down to 0.3 Tc, but atlower temperatures changes in the twisted cluster propagation appear tooccur, to maintain stability and polarization. These have not been ade-quately studied and may lead to revisions of the current model at the verylowest temperatures.

3.5 Analytical model of turbulent front

The measured front velocity in Figure 17 displays below 0.4 Tc twoplateaus which are separated by a transition centred at 0.25 Tc. The numer-ical analysis in Section 3.4 places this transition in the temperature regionwhere the Kelvin-wave cascade acquires growing importance, that is, inthe regime where subinter-vortex scales start to influence energy trans-fer. The peculiar shape of the measured vf(T) curve prompted L’vovet al. (2007a) to examine energy transfer from the quasi-classical lengthscales (where bundles of vortices form eddies of varying size with a Kol-mogorov spectrum) to the quantum regime (where Kelvin-waves expandon individual vortex lines). The result is a bottleneck model (L’vov et al.,2008) which matches energy transfer across the crossover region fromsuperinter-vortex length scales to subinter-vortex scales. It can be fittedwith reasonable parameters to the measured vf(T) curve (Eltsov et al.,2007).

3.5.1 Two-Fluid Coarse-Grained Equations

To describe the superfluid motions on length scales exceeding the inter-vortex distance �, we make use of the coarse-grained hydrodynamicequation for the superfluid component. This equation is obtained fromthe Euler equation for the superfluid velocity U ≡ Us, after averaging overthe vortex lines which are assumed to be locally approximately aligned,forming vortex bundles which mimic the eddies in viscous flow (see thereview by Sonin, 1987),

∂U∂t

+ (U · ∇)U + ∇m = fmf . (13)


Here, m is the chemical potential and fmf the mutual friction force(Equation (3)), which can be rewritten as follows:

fmf = −α′(U − Un) × ω + α ω × [ω × (U − Un)]. (14)

We use ω = ∇ × U for the superfluid vorticity; ω ≡ ω/ω is the unit vectorin the direction of the mean vorticity; Un is the velocity of the normalcomponent (which is fixed). In flow with locally roughly aligned vortices,the mutual friction parameters define the reactive (∝ α′) and dissipative(∝ α) forces acting on a bunch of vortex lines as it moves with respect tothe normal component in a field of slowly varying vortex orientation.

We shall work in the rotating reference frame where Un = 0. In thisframe, the reactive first term in Equation (14) renormalizes the inertialterm U × ω on the left-hand side (LHS) of Equation (13), introducing thefactor 1 − α′ (Finne et al., 2003). The relative magnitude of the two nonlin-ear terms, the ratio of the inertial term and of the friction term (the latteris the second term on the right-hand side RHS of Equation (14)), is theReynolds number in this hydrodynamics. It proves to be the flow veloc-ity independent ratio of the dimensionless mutual friction parameters:ζ = (1 − α′)/α (which was introduced by Finne et al., 2003 in the formq = 1/ζ). To arrive at a qualitative description of the front, we follow L’vovet al. (2004) and simplify the vectorial structure of the dissipation term andaverage Equation (14) over the directions of the vorticity ω (at fixed direc-tion of the applied counterflow velocity v). With the same level of accuracy,omitting a factor 2/3 in the result, we get

fmf ⇒ ⟨fmf⟩ω/|ω| = −2

3αωeff v ⇒ −αωeff v , (15)

ωeff ≡√⟨|ω|2⟩ .

Here and henceforth, 〈. . .〉 denotes ‘ensemble averaging’ in the propertheoretical meaning. From the experimental point of view, 〈. . .〉 can beconsidered as time averaging in a frame which moves with the front(over a time interval, during which the front propagates a distanceequal to its width). The resulting mean values, to be discussed in Equa-tions (16) and (19), can be considered as functions of (slow) time andspace. We follow L’vov et al. (2004) and neglect the fluctuating turbu-lent part of ωeff in Equation (15). We thus replace |ω| by its mean value,the vorticity from rotation. In this approximation, Equation (15) takes thesimple form:

fmf = −�U , � ≡ αωeff. (16)


3.5.2 Flow Geometry, Boundary Conditions and ReynoldsDecomposition

Axial flow geometry and flat idealisation. As a simplified model of therotating cylinder, we consider first the flat geometry with x as the stream-wise, y as the cross-stream and z as the front-normal directions. A morerealistic axial symmetry will be discussed later. We denote the position ofthe front with Z(t) and look for a stationary state of front propagation atconstant velocity −Vf:

Z(t) = −V f t. (17)

The normal velocity is zero everywhere, Un(r, t) = 0, which corre-sponds to co-rotation with the rotating cylinder. The vortex front prop-agates in the region z < Z(t), where the superfluid velocity is −V∞, whilefar away behind the front at z > Z(t), the superfluid velocity U tends tozero (i.e., towards the same value as the normal fluid velocity).

Reynolds decomposition and definitions of the model. Following thecustomary tradition (see, Pope, 2000), we decompose the total velocityfield U into its mean part V and the turbulent fluctuations v with zeromean:

U = V + v, V = 〈U〉 , 〈v〉 = 0. (18a)

In our model,

V = x V, V ≈ � r. (18b)

The following mean values are needed: the mean velocity shear S(r, t), theturbulent kinetic energy density (per unit mass) K(r, t) and the Reynoldsstress W(r, t). Their definitions are as follows:

S(r, t) ≡ ∂V∂z

, K(r, t) ≡ 12

⟨|v|2

⟩, W(r, t) ≡ − 〈vxvz〉. (19a)

Also we will be using the kinetic energy density of the mean flow

KV(r, t) ≡ 12

[V(r, t)]2 (19b)

and the total density of the kinetic energy

K(r, t) ≡ KV(r, t) + K(r, t) (19c)


Clearly, the total kinetic energy in the system, E(t), is:

E(t) =∫

K(r, t)dr. (20)

3.5.3 Balance of Mechanical Momentum and Kinetic Energy

Balance of mechanical momentum. Averaging Equation (13), with thedissipation term in the form of Equation (16), one gets for the planargeometry:

∂V∂t

− ∂W∂z

+ αωeff V = 0. (21)

Mean-flow kinetic energy balance. Multiplying Equation (21) with V, onegets the balance for the kinetic energy of the mean flow, KV, defined inEquation (19b):

∂KV

∂t− ∂

∂z[VW] + S W + αωeff KV = 0. (22)

The second term on the LHS of this equation describes the spatialenergy flux and does not contribute to the global energy balance ofthe entire rotating cylinder. The next term, S W, is responsible for theenergy transfer from the mean flow to the turbulent subsystem; thisenergy finally dissipates into heat. The last term, αωeff KV, representsdirect dissipation of the mean flow kinetic energy into heat via mutualfriction.

Turbulent kinetic energy balance. Let us take Equation (13) for the velocityfluctuations, with the dissipation term in the form Equation (16), and weget an equation for the energy balance which involves triple correlationfunctions. It describes the energy flux in (physical) space. In the theoryof wall bounded turbulence (Pope, (2000)), these triple correlations aretraditionally approximated with second-order correlation functions. Formore details about the closure procedures we refer to L’vov et al. (2006a,2007b) and references cited therein.

To be brief, notice that the total rate of kinetic energy dissipation in thevortex front has two contributions in well-developed turbulence,

εdiss = εdiss,1 + εdiss,2. (23a)


The first term, εdiss,1, arises from mutual friction which acts on theglobal scale. It can be estimated from Equation (13) as

εdiss,1 � αωeffK(z, r), (23b)

where K(z, r) is the turbulent kinetic energy from Equation (19a). Thesecond contribution, εdiss,2, originates from the energy cascade towardssmall scales, where the actual dissipation occurs. In well-developed tur-bulence of viscous normal fluids, this dissipation is caused by viscosity.It dominates at the smallest length scales known as the Kolmogorovmicroscope. In superfluid turbulence, the viscous contribution is absent.Instead, at moderate temperatures, it is replaced by mutual friction.When mutual friction becomes negligibly small at the lowest temper-atures, the turbulent energy, while cascading down to smaller lengthscales, is accumulated into Kelvin-waves at some crossover scale. Ulti-mately the energy then cascades further down where it can dissipate,for example, by the emission of excitations (phonos in 4He II and quasi-particles in 3He-B). In any case, the energy has to be delivered to smalllength scale motions owing to the nonlinearity of Equation (13). Clearlyin steady state conditions, the rate of energy dissipation at small scales,εdiss,2, is equal to the energy flux, εflux, from the largest scales, where theenergy is pumped into the system (from the mean flow via the shear S,as follows from Equation (22)). The nonlinearity of Equation (13) is ofstandard hydrodynamic form, and the associated energy flux can be esti-mated by dimensional reasoning, as suggested by Kolmogorov in 1941(see Pope, 2000):

εdiss,2 = εflux � b K3/2(r)/L(r). (23c)

Here b is a dimensionless parameter, which in the case of classical (vis-cous) wall-bounded turbulence can be estimated as bcl � 0.27 (see L’vovet al., 2006a). In Equation (23c), L(r) is the outer scale of turbulence (whichdefines the length scale of the largest eddies containing the main part ofthe turbulent kinetic energy). Clearly near the centreline of the cylinder,L(r) is determined by the thickness �(r) of the turbulent front at givenradius r: L(r) � �(r). Near the wall of the cylinder, the size of the largesteddies is limited by the distance to the wall, R − r. Therefore in this region,L � R − r. In the whole cylinder, L should be the smaller of these twoscales so that one can use an interpolation formula

L−1(z) = �(r)−1 + (R − r)−1. (23d)


The resulting energy balance equation, which accounts for the energydissipation in Equations (23a–23d), can be written (in cylindrical coordi-nates r, z and ϕ) as follows:

∂K∂t

+ αωeff K + b K3/2

L

−g[ ∂

∂zL√

K∂

∂zK + 1

r∂

∂rr L√

K∂

∂rK]

= S W.

(24)

Here K, W and S are defined in Equation (19a). Generally speaking, theyare functions of r, z and time t. The outer scale of turbulence L dependson r and t in our approximation (Equation (23d)).

The RHS of Equation (24) represents the energy flux (per unit mass)from the mean flow to the turbulent subsystem. This expression rigor-ously follows from Equation (13) and is exact. The two terms on the LHSof Equation (24) (proportional to α and b) describe the energy dissipa-tion (Equation (23)). The last two terms on the RHS of Equation (24) areproportional to the phenomenological dimensionless parameter g ≈ 0.25,which was estimated by L’vov et al. (2007b). These terms model turbulentdiffusion processes (in the differential approximation) with the effectiveturbulent diffusion parameter L g

√K, now expressed in cylindrical coor-

dinates. The first term in parenthesis describes turbulent diffusion in thedirection normal to the front, which is the main reason for the propagationof the turbulent front. The second term, which vanishes in the flat geome-try, describes the turbulent energy flux in the radial direction towards thecentreline of the sample cylinder. This term plays an important role in thepropagation of the front as a whole. The reason is that the central part ofthe front, where the direct conversion of the mean flow energy to turbu-lence is small, does not contain enough energy to exist without the radialflux of energy.

Total kinetic energy balance. Adding ∂KV/∂t from Equation (22) and ∂K/∂tfrom Equation (24), one can see that the term S W, responsible for thetransfer of energy from the mean flow to the turbulent flow, cancels andwe arrive at a balance equation for K ≡ KV + K:

∂K∂t

+ αωeff K + b K3/2

L − ∂

∂z[VW]

−g[ ∂

∂zL√

K∂

∂zK + 1

r∂

∂rr L√

K∂

∂rK]

= 0.

(25a)


Integrating this relation over the cylinder, we see that the energy diffusionterms do not contribute to the total energy balance:

E(t) ≡∫

K(r, t)dr, (25b)

dEdt

= −∫

dr[αωeff K + b K3/2

L]. (25c)

To summarise this section, we note that Equations (21) and (24) allowus, at least in principle, (i) to describe the propagating turbulent front in arotating superfluid, (ii) to find the front velocity Vf and (iii) to describe thestructure of the front: its effective width �(r), the profiles for the meanshear S(r, z) the Reynolds stress W(r, z) and the kinetic energy K(r, z).Here we present only some preliminary steps in this direction, based ona qualitative analysis in Section 3.5.4 of the global energy balance (Equa-tion (25a)), and discuss the role of the radial turbulent diffusion of energyin Section 3.5.5. In Section 3.5.6, we finally specify the model, account-ing explicitly for the bottleneck at the classical-quantum crossover, asdescribed by L’vov et al. (2007a), and explain how it is influenced by thetemperature-dependent mutual friction.

3.5.4 Qualitative Analysis of Global Energy Balance

The total kinetic energy E is dissipated by the propagation of the turbulentfront at constant velocity Vf, as expressed in Equation (8). This meansthat during time t the length of the cylinder with unperturbed mean flowV = � r decreases by Vf t. Using Equation (8), we can write the overallenergy budget (Equation (25c)) as:

Vf = 4π�2R4

∫dr[αωeff K(z, r) + b K3/2(z, r)

L(r)

]. (26a)

This equation requires some corrections, because in its derivation we werea bit sloppy. First of all, approximation (Equation (16)) for the mutualfriction force fmf cannot be applied for the analysis of laminar flow, wherethe actual orientations of vorticity and velocity are important. To fix this,we present the RHS of Equation (26a) as a sum of two contributions,

Vf = Vf,lam + Vf,turb , (26b)

and for Vf,lam, we use Equation (12). To get the corrected equation forVf,turb, we replace in Equation (26a) K with its turbulent part K andreplace b with b ≡ b(1 − α′) in order to account for the correction to the


nonlinear term in Equation (13) from the reactive part of the mutual fric-tion (Equation (14)) (proportional to α′). Integrating the result over theazimuthal angle ϕ and making use of the axial symmetry of the problem,we get

Vf,turb = 8�2R4

∫ R−�

0r dr dz

[αωeff K(z, r) + b K3/2(z, r)

L(r)

]. (26c)

The hydrodynamic equation (Equation (13)) is not applicable near the wall,where R − r is less than the mean inter-vortex distance �. Therefore, thisregion is excluded from the integration (Equation (26c)).

In the limit of a fully-developed turbulent boundary layer (TBL), theturbulent kinetic energy and Reynolds stress are independent of the dis-tance in the z direction to the boundary of turbulence. Here we consider theturbulence in the front to be bounded in the z direction, that is, we assumethat the front thickness �(r) < R and ignore the influence of the cylindri-cal container wall. Therefore, it is reasonable in the qualitative analysis ofEquation (26c) to ignore the z dependence of K(z, r) and W(z, r), replacingthese objects with their mean values across the TBL:

K(r, z) ⇒ K(r) , W(r, z) ⇒ W(r) , ωeff(r, z) ⇒ ωeff(r). (26d)

As seen from Equation (26d), we use the same approximation also for ωeff.Now we can trivially integrate Equation (26c) with respect to z:

Vf,turb = 8�2R4

∫ R−�

0

[αωeff(r)K(r) + b K

3/2(r)

L(r)

]�(r)r dr. (26e)

Here �(r) is the characteristic width of the TBL at the distance r fromthe central axial. To perform the next integration over r, one needs toknow the r-dependence of �, ωeff and K. Notice first that �(r)ωeff(r) hasthe dimension of velocity. Therefore, one expects that in the self-similarregime of a fully-developed TBL, �(r)ωeff(r) has to be proportional tothe characteristic velocity, which is � r. In other words, we expect that�(r)ωeff(r) ∝ r. As we will see below in Section 3.5.5, �(r) ∝ r while ωeffis r-independent. With the same kind of reasoning, we can conclude thatK and W have to be proportional to r2 since they have the dimensionalityof velocity squared:

�(r)ωeff = a�r , K(r) = c2(�r)2 , W(r) = c (�r)2. (27)


Now integrating (Equation (26e)), one gets

vf ≡ Vf

�R� (2 c)3/2b (1 − α′)A(R/�) + 4 c α

5 a, (28a)

A(R/�) = 0.2 + d[ ln(R/�) − 137/60 + 5�/R + . . . ], (28b)

where we used d = �(r)/r. At � = 1 rad/s, the ratio R/� ≈ 17. Thisgives A(R/�) ≈ A(17) ≈ 1.8. We take b = bcl and choose the parametersa = 0.2, c = 0.25 and d = 2 to fit the measurement in the region (0.3 –0.4) Tc. With these parameters, Equation (28) gives vf ≈ 0.16 in the limitT → 0 (when α = α′ = 0) and a very weak temperature dependence up toT � 0.45 Tc.

3.5.5 Role of the Radial Turbulent Diffusion of Energy

According to experimental observations, in steady state, the front propa-gates as a whole with the velocity vf, independent of the radial position r.To make this possible, the turbulent energy has to flow from the near-wallregion, where the azimuthal mean velocity and consequently the energyinflux into the turbulent subsystem are large, towards the centre, wherethe influx goes to zero.

To clarify the role of the radial energy flux, consider Equation (24)averaged in the z direction:

αωeff K(r) + b [K(r)]3/2

L(r)− g

r∂

∂rr L(r)

√K(r)

∂

∂rK(r) = � r

�(r)W(r).

(29)On the RHS of this equation we replaced S(r) with its natural esti-mate � r/�(r). According to Equation (23d), close to the cylinder axis,L(r) ≈ �(r). In this region, Equation (29) has a self-similar scaling solu-tion (Equation (27)), in which ωeff is indeed r-independent, �(r) ∝ r,and for the Reynolds-stress constant c in Equation (27) one finds therelationship

cc

= αωeff

�

�(R)

R+ b

2

√c2

+ 2 d√

2 c[�(R)

R

]2. (30a)

In the vicinity of the wall, where R − r � R, the outer scale of turbulence isL ≈ R − r and goes to zero when r → R. To account for the influence of thecylindrical wall in this region, we assume that, similar to a classical TBL,the two pair velocity correlations W(r) and K(r)have the same dependenceon distance from the wall. Therefore, their ratio is approximately constant:


W(r)/K(r) ≈ const. If so, the energy balance Equation (24) dictates

W(r) ∼ K(r) ∼ (R − r)2, (30b)

with some relation between constants, similar to Equation (30a). Actually,when R − r becomes smaller than the inter-vortex distance �, the wholehydrodynamic approach (Equation (13)) together with Equation (29) fails.Therefore, we cannot expect that W(r) and K(r) really go to zero whenr → R. Rather they should approach some constant values dominated bythe vortex dynamics at distances R − r � �.

To summarise, we should say that Equation (29) and its solution (Equa-tion (30)) cannot be taken literally as a rigorous result. They give aqualitative description of the kinetic energy profiles. Indeed, the predictedprofiles are in qualitative agreement with numerical simulation calcula-tions, as seen in Figure 23: here K(r) goes to zero when r → 0, increasesat small r approximately as r2, as concluded in Equations (27) and (30a),reaches a maximum at r/R � 1 and then decreases in qualitative agreementwith Equation (30b). Similar comparison with other numerical results inSection 3.4 makes it believable that the analysis of the global energy bal-ance in Section 3.5.4 and the predicted plateau for vf in the T → 0 limit arereasonable.

Nevertheless, both in the region of lower and higher temperatures, theexperiment shows deviation from this ‘plateau’ (Figure 26). The reasonfor this deviation at T > 0.35 Tc is that turbulence is not well developednear the cylinder axis where the shear of the mean velocity, responsi-ble for the turbulent excitation, decreases. Therefore, in the intermediatetemperature region only part of the front volume is turbulent. Whenthe temperature decreases, the turbulence expands towards the axis. Weshould therefore account also for the laminar contribution to the frontvelocity (Equation (12)) near the axis. Notice furthermore that the simplesum of laminar and turbulent contributions to Vf in Equation (26b) over-simplifies the situation, by not accounting for the turbulent motions inEquation (26c). We will not discuss this issue but just suggest an interpo-lating formula between the laminar and turbulent regimes, which has ashorter intermediate region than Equation (26b):

vf =√

v2f, lam + v2

f, turb , (31)

where vf, lam and vf, turb are given by Equations (12) and (28). This interpo-lation is shown in Figure 26 as a thin line for T < 0.3 Tc and as a thick linefor T > 0.3 Tc. The agreement with measurements above 0.3 Tc is good,but there is a clear deviation below 0.25 Tc, where α � 10−2. Below an


0.2 0.3 0.4 0.5T/Tc

T/Tc

Vf/

(ΩR

)

0

0.1

0.2

0.3

0.4

Quantum

Quasi-classical

0 0.2 0.4 0.60

0.1

0.2

b bcl

FIGURE 26 Scaled front velocity versus temperature, similar to Figure 17 butemphasising the comparison of measurements in Section 3.3 to analytical results inSection 3.5 in the turbulent regime at low temperatures. The thin and thick solid linesshow consecutive model approximations which sequentially account for dissipation inturbulent energy transfer Equation (28b) and bottleneck effect in Section 3.5.6. Insert:value of parameter b(T ) in Equation (28), which was used for the thick solid line in themain panel.

explanation is outlined which takes into account the quantum characterof turbulence, since as shown by Volovik (2003), individual vortex linesbecome important below 0.3 Tc in the region where the transition to thelower plateau occurs in Figure 26.

To appreciate the last aspect, note that the mean free path of 3He quasi-particles at the conditions of the measurements at 29 bar pressure andT � 0.3 Tc is close to �, while at 0.2 Tc it exceeds R. This change fromthe hydrodynamic to the ballistic regime in the normal component mayinfluence the mutual friction force acting on individual vortices. However,what is important for the interpretation of the measurements is that theeffect of the normal component on the superfluid component becomesnegligibly small. Therefore, it actually does not matter what the physicalmechanism of this interaction is: ballistic propagation of thermal excitationwith scattering on the wall or mutual friction that can only be describedin the continuous-media approximation, as given by Equation (14). Forsimplicity, we use the hydrodynamic approximation (Equation (14)) inSection 3.5.6 to describe the turbulent kinetic energy dissipation at lowtemperatures.


3.5.6 Mutual Friction and Bottleneck Crossover from Classical toQuantum Cascade

Bottleneck at zero temperature. At low temperatures, when mutual fric-tion becomes sufficiently small, the energy flux towards small length scales(or large wave vectors k), εk , propagates down to the quantum scale �,and vortex discreteness and quantisation effects become most important.Even though some part of the energy is lost in intermittent vortex recon-nections, the dominant part proceeds to cascade below the scale � bymeans of nonlinearly interacting Kelvin-waves (see Kozik and Svistunov(2008a,b); Vinen et al. (2003) and references there). The Kelvin-waves aregenerated by both slow vortex filament motions and fast vortex reconnec-tion events. As shown by L’vov et al. (2007a), the important point for therate of energy dissipation (and consequently for the turbulent front veloc-ity) is that Kelvin-waves are much less efficient in the downscale energytransfer than classical hydrodynamic turbulence: in order to provide thesame energy flux as in the hydrodynamic regime, the energy density ofKelvin-waves at the crossover scale � has to be �10/3 times larger thanthat of hydrodynamic motions. For 3He-B, � ≡ ln(�/a0) � 10, where a0 isthe vortex-core radius. Assuming that the energy spectrum is continuousat the crossover scale � and that no other mechanisms intervene betweenthe classical and Kelvin-wave cascades, then to maintain the same valueof energy flux, there must be a bottleneck pile-up of the classical spec-trum near this scale by the factor �10/3. To account for this phenomenonof energy pileup, we construct the ‘warm cascade’ solution which will bedescribed in what follows.

For a qualitative description, we first define the hydrodynamic kineticenergy density (in the 1D k-space), Ek , related to the total kinetic energy Eas follows:

E ≡∫

Ek dk. (32a)

The Kolmogorov-1941 cascade of hydrodynamic turbulence (with k-independent energy flux, εk ⇒ ε=const) is described with the spectrum:

EK41

k � ε2/3|k|−5/3. (32b)

The energy flux carried by the classical hydrodynamic turbulence withthe K41 spectrum (32b) cannot adequately propagate across the crossoverregion at �. Therefore, hydrodynamic motions on larger length scales(smaller wave-vectors) will have increased energy content up to the levelE1/�, as required when the same energy flux has to be maintained by


means of Kelvin-waves. As a result, for k ≤ 1/�, the spectrum of hydro-dynamic turbulence EHD

k will not have the K41 scale-invariant form EK41

kgiven by Equation (32b). To get a qualitative understanding of the result-ing bottleneck effect, we use the so-called warm cascade solutions foundby Connaughton and Nazarenko (2004). These solutions follow from theLeith (1967) differential model for the energy flux of hydrodynamicalturbulence,

εk = −18

√|k|13Fk

dFk

dk, Fk ≡ EHD

k

k2 , (33)

where Fk is the 3-dimensional (3D) spectrum of turbulence. The genericspectrum with a constant energy flux can be found as the solution to theequation εk = ε:

Fk =[ 24ε

11|k|11/2 +( Tπρ

)3/2]2/3. (34)

Here the range of large k values belongs to the thermalised part ofthe spectrum, with equipartition of energy characterised by an effec-tive temperature T, namely T/2 of energy per degree of freedom, thus,Fk = T

/πρ and Ek = Tk2/πρ. At low k, Equation (34) coincides with the

K41 spectrum (Equation (32b)).This ‘warm cascade’ solution describes reflection of the K41 cascade

and stagnation of the spectrum near the bottleneck scale which, in ourcase, corresponds to the classical-quantum crossover scale. To obtain thespectrum in the classical range of scales, it remains to find T by matchingEquation (34) with the value of the Kelvin-wave spectrum at the crossoverscale Ek ∼ κ2/�. This gives T/ρ ∼ κ2� ∼ (κ11/�5ε)1/4.

Obviously, the transition between the classical and quantum regimes isnot sharp, and in reality we should expect a gradual increase of the role ofthe self-induced wave-like motions of individual vortex lines with respectto the collective classical eddy type of motions of vortex bundles. Thus,the high-wave number part of the thermalised range is likely to be waverather than eddy dominated. However, the energy spectrum for this partshould still be of the same k2 form which corresponds to equipartition ofthermal energy. This picture, as explained below, relies on the assumptionthat the self-induced wave motions have small amplitudes and, therefore,do not lead to reconnections.

The resulting spectrum, including its classical, quantum and crossoverparts, is shown in Figure 27 as a log-log plot. It is important to note that inthe quantum range k > 1/�, in addition to the cascading energy associatedwith Kelvin-waves, there is also energy associated with the tangle of vortex


Ene

rgy

dens

ityε k

10221022

102

103

1021

10

1

1021

k ¯

k 25/3

k 27/5

k 21

k 2

10 1021

FIGURE 27 Energy spectra Ek in the classical, k < 1/�, and quantum, k > 1/�, rangesof scales. The two straight lines in the classical range indicate the pure K41 scalingEK41k ∝ k−5/3 of Equation (32b) and the pure thermodynamic scaling Ek ∝ k2. In the

quantum range, the solid line indicates the Kelvin-wave cascade spectrum ∝ k−7/5,whereas the dash-dotted line marks the spectrum corresponding to the non-cascadingpart of the vortex tangle energy ∝ k−1.

filaments (shown in Figure 27 with the dash-dotted line). The energy spec-trum of this part is ∼|k|−1, which is simply the spectrum associated with asingular distribution of vorticity along 1D curves in 3D space (Araki et al.,2002). It does not support a downscale cascade of energy. The cascadingand noncascading parts have similar energies at the crossover scale, thatis, the wave period and the amplitude are of the order of the characteristictime and size of evolving background filaments. In other words, the scalesof the waves and of the vortex ‘skeleton’ are not separated enough to treatthem as independent components. This justifies matching the classicalspectrum at the crossover scale with the Kelvin-wave part alone, ignoringthe vortex skeleton. This is valid up to an order-one factor and justifies theway of connecting the skeleton spacing � to the cascade rate ε.

Effect of mutual friction on the bottleneck crossover. Returning back tothe propagating turbulent vortex front, recall that in the measurementsof Section 3.3 the inertial interval R/� is about one decade. Therefore, thedistortion of the energy spectrum owing to the bottleneck reaches the outerscale of turbulence. This leads to an essential suppression of the energyflux at any given turbulent energy or, in other words, to a decrease in the


effective parameter b, which relates εand K in the estimate (Equation (23c)).The effect is more pronounced at low temperatures when mutual friction issmall; thus b(T) should decrease with temperature. We analyse this effectwith the help of the stationary energy balance equation for the energyspectrum Ek in k-space:

dεk

dk= −�(T)Ek , εk ≡ −(1 − α′)

√k11Ek

d(Ek/k2)

8 dk, (35)

where �(T) is the temperature-dependent damping in Equation (16), andthe energy flux over scales ε(k) is taken in the Leith (1967) differentialapproximation. In addition to Equation (33), we included in Equation (35)the mutual friction correction factor (1 − α′) and substituted Fk = Ek/k2.

Figure 28 displays the set of solutions for Equation (35). We useL/� = 12 as the ratio of the outer and crossover scales and characterisethe bottleneck with the boundary condition Ek/[k3d(Ek/k2)/dk] = −4 · 105

at the crossover scale. One goal of these calculations is to find the slopeof the function Fk at the beginning of the inertial interval k = k+ ≡ 2π/Lwhich, according to Equation (35), characterises the rate of energy inputinto the system at fixed value of the total energy E , that is, the phe-nomenological parameter b in the estimate (Equation (23c)). The dashedline in the right panel shows the Kolmogorov value of this slope, −11/3,which is associated with the classical value of b = bcl ≈ 0.27. Thus theratio 3bcl|F′|/11 can be interpreted as the effective value of b for a givenvalue of damping �. To relate � with the temperature-dependent value

1 1.5 2 5 10x

0.01

0.1

1

10

100

x2 F

25/3

2

0.001

0.01

g50

0.1

0.5

1

11025 1024 1023 1022 1021

g

1.5

2

2.5

3

3.5

4

4.5

2F

′(0)

3 7

FIGURE 28 On the left, the solutions of the differential Equation (35) for different

values of the dimensionless mutual friction parameter γ ≡ �/

√k3+Ek+ have been

plotted. The smallest wave vector k+ corresponds to the outer scale of turbulence:L ≈ 2π/k+. The wave vectors are normalised with respect to k+: x ≡ k/k+ andF (x) ≡ Fx k+/Fk+ . The dashed lines denote slopes of Ek with K41 scaling Ek ∝ k−5/3

and thermodynamic scaling Ek ∝ k2. On the right, the slope F ′ ≡ dF (x)/dx is shown atx = 1 for different values of mutual friction γ . The horizontal dashed line marks theK41 value of slope, 11/3.


α(T), we substitute the r independent ωeff from Equation (27) to Equa-tion (16). This gives � = α (a/d)�. The resulting function b(T), shownin the inset of Figure 26 (for � = 1 rad/s), decreases from its classicalvalue bcl ≈ 0.27 to � 0.1 at T < 0.2 Tc. Now, after accounting for the tem-perature dependence of b(T) in Equation (28), we get the temperaturedependence for the propagation velocity vf of the quantum-turbulentfront shown in Figure 26 by the bold solid line below 0.3 Tc. This fit isin good agreement with the measured data. We thus have to conclude thatin this particular measurement the rapid drop in the dissipation rate onentering the quantum regime can be explained as a consequence of the rel-atively close proximity of the outer and quantum crossover scales in thismeasuring setup.

3.6 Summary: Turbulent Vortex Front Propagation

In 3He-B, strong turbulence is restricted to the lowest temperatures below0.4 Tc. Depending on the type of flow, turbulence varies in form and losses.The usual reference point is an isotropic and homogeneous turbulent vor-tex tangle. Here we have discussed an opposite extreme, the conversion ofmetastable rotating vortex-free counterflow to the stable equilibrium vor-tex state. The task is to explain how increasing turbulence with decreasingtemperature influences this type of polarized vortex motion in steady statepropagation. The deterministic part of this motion takes place in the formof a spiraling vortex front followed by a helically twisted vortex clus-ter. When mutual friction decreases, turbulent losses start to contributeto dissipation, concentrating in the propagating front and immediatelybehind it.

Experimentally, the conversion of a metastable vortex-free state to anequilibrium array of rectilinear vortex lines can be arranged to occur at con-stant externally adjusted conditions. This is done with different types ofinjection techniques, which trigger the formation of a propagating vortexfront and the trailing twisted cluster behind it. In a long rotating column,the propagation can be studied in steady state conditions. At tempera-tures above 0.4 Tc, the propagation is laminar, but below 0.4 Tc a growinginfluence from turbulence appears. This is concluded from the measuredpropagation velocity which provides a measure of the dissipation in vortexmotion. In addition to the increasing turbulent dissipation below 0.4 Tc,the measurement shows a peculiar transition at ∼0.25 Tc between twoplateaus, and a temperature-independent finite value of dissipation onapproaching the T → 0 limit.

So far measurements on vortex propagation exist only for a columnwith circular cross section. This is an exceptional case of high stability (ifthe cylindrical symmetry axis and the rotation axis are sufficiently wellaligned). Here the crossover from the laminar to the turbulent regime is


smooth as a function of mutual friction dissipation. The same smoothbehavior is also seen in the response to spin down, after a step-like stopof rotation. This is in stark contrast to the spin down of a column withrectangular or square cross section, as we will see in Section 4. Similar dif-ferences seem to apply in viscous hydrodynamics where flow in a circularpipe is believed to be asymptotically linearly stable for all Reynolds num-bers, in contrast to a pipe with square cross section (Peixinho and Mullin,2006). In these measurements the formation and decay of turbulent plugsis monitored. Vortex plugs and fronts have also been observed in pipe flowof superfluid 4He through long circular capillaries (Marees et al., 1987).

Numerical vortex dynamics calculations have been used to examinethe turbulent contributions to dissipation in vortex front propagation.This analysis demonstrates the increasing role of turbulent excitations onsubinter-vortex scales with decreasing temperature. Analytical argumentshave been developed which explain that the transition to the lower plateauin the measured overall dissipation is caused by the difficulty to bridge theenergy cascade from the quasi-classical to the quantum regime as a func-tion of decreasing mutual friction dissipation. This bottleneck scenario ofL’vov et al. (2007a) in energy transfer was thus directly inspired by theexperimental result. It is expected to apply foremost to polarised vortexmotion in the rotating column where vortex reconnections are suppressed.This situation differs from the case of an ideal homogeneous vortex tangle,where Kozik and Svistunov (2008a,b) suggest that a bottleneck is avoidedowing to the high reconnection rate.

As for the second leveling off at the lowest temperatures, at presenttime three different types of measurements (Bradley et al., 2006; Eltsovet al., 2007; Walmsley et al., 2007a) conclude that dissipation in vortexmotion remains finite at the lowest temperatures. This applies for boththe fermion superfluid 3He-B and the boson case of 4He. Although themechanisms for dissipation in the T → 0 limit in these two superfluidsare different and still under discussion, the results suggest that coherentquantum systems can be inherently lossy even in the T = 0 state.

4. DECAY OF HOMOGENEOUS TURBULENCE INSUPERFLUID 4HE

4.1 Introduction and Experimental Details

4.1.1 Quasiclassical and Ultraquantum Types of Superfluid Turbulence

In this section, we switch to nearly homogeneous and isotropic superfluidturbulence with an emphasis on the recent experiments in Manchester.A truly homogeneous turbulence is of course impossible to achieve in real


experiments because the tangle is always confined by the container wallsand can have other inhomogeneities specific to the particular process ofits generation.

The modern paradigm of homogeneous isotropic turbulence at highReynolds numbers Re in classical liquids is that there is a broad range ofwave numbers k within which the energy, pumped at large scales, getscontinuously redistributed without loss towards smaller length scales,thus arranging a steady-state distribution in k-space of the Kolmogorovtype (Batchelor, 1953; Frisch, 1995; Kolmogorov, 1941a,b) (Equation (32b)):

EK41

k = Cε2/3|k|−5/3, (36)

where the Kolmogorov constant was found to be C ≈ 1.5 (Sreenivasan,1995). In the steady state, the energy flux ε is equal to the dissipation of thekinetic energy through the shear strain in the flow at short length scales;its rate integrates to

E = −νclω2cl, (37)

where νcl is the kinematic viscosity and ωcl is the r.m.s. vorticity. In real-istic systems, the distribution Equation 36 is truncated at small k = k1 byeither the length scale of forcing or the container size h, and at large k = k2

by the Kolmogorov dissipation scale λcl(Re) ∼ k−12 which decreases with

increasing Re = (k2/k1)4/3. In superfluids, the inertial cascade is expected

to operate if the mutual friction parameter α is sufficiently small (Vinen,2000). At large length scales > �, the cascade is classical. However, at shortlength scales < �, the cascade becomes quantum as the discreteness of thevorticity in superfluids adds new behaviour. At sufficiently low tempera-tures, when the energy reaches very small scales � � without dissipation,it is expected that the quantum cascade takes the form of a nonlinearcascade of Kelvin-waves (Kozik and Svistunov, 2004; Svistunov, 1995),eventually being truncated at some quantum dissipation scaleλq(α). Whilethe theory of this regime at very high k is now established, no directobservations exist so far. The most complicated is, of course, the tran-sitional region between these two clear-cut limits, the Kolmogorov andKelvin-wave cascades. The question being currently debated is whetherthe energy stagnates at k < �−1 due to the poor matching in the kinetictimes of the two cascades (L’vov et al., 2007a), or gets efficiently convertedfrom 3D classical eddies to 1D waves along quantised vortex lines withthe help of various reconnection processes (Kozik and Svistunov, 2008a).

When comparing the two superfluid isotopes from the experimentalpoint of view, the T = 0 limit in superfluid 4He is achievable at someT < 0.5 K (Vinen, 2000; Walmsley et al., 2007a) while for 3He-B much lower


temperatures T < 0.5 mK are required. Owing to the two or three orders ofmagnitude smaller core size in 4He, a0 ∼ 0.1 nm, the Kelvin-wave-cascadeis expected to extend over a broader range of length scales, down to∼3 nm (Kozik and Svistunov, 2008b; Vinen, 2001). It has been suggested(Vinen and Niemela, 2002; Vinen, 2002) that in 3He-B Kelvin-waves shouldbecome overdamped at frequencies ∼10 kHz (corresponding to wave-lengths∼2 mm) due to the resonant scattering on core-bound quasiparticles(Kopnin and Salomaa, 1991)).

Flow on a scale greater than the inter-vortex distance �, which is initiallytypically between ∼10 and 100 mm in experiments, can be obtained bymechanical stirring of the liquid. So far the following methods have beenused: counter-rotating agitators with blades (Maurer and Tabeling, 1998),pipe flow (Roche et al., 2007; Smith et al., 1999), flow through an orifice(Guenin and Hess, 1978), towed grids (Niemela et al., 2005; Smith et al.,1993; Stalp et al., 1999), vibrating grids (Bradley et al., 2006; Davis et al.,2000; Fisher and Pickett, 2008; Hänninen et al., 2007a; Vinen and Skrbek,2008), as well as wires (Bradley, 2000; Nichol et al., 2004.) and microspheres(Schoepe, 2004), plus most recently impulsive spin-down to rest (Walmsleyet al., 2007a) of a rotating container. It is also possible to initiate large-scale flow in superfluid 4He without any moving parts or rotation of thecryostat: by either running thermal counterflow in wide channels at T >

1 K (Barenghi and Skrbek, 2007; Chagovets et al., 2007) or a jet of injectedcurrent (Walmsley and Golov, 2008a).

Actually, the turbulence can take two very different forms dependingon whether the forcing is at scales above or below �. So far we were dis-cussing the flow on classical scales> �, where the energy cascades towardsshorter length scales like in the Richardson cascade in classical turbulence;the large quasiclassical eddies being the result of correlations in polarisa-tion of vortex lines. On the other hand, when forced on quantum scales< �, the resulting uncorrelated tangle has no classical analogs and shouldhave completely different dynamics first described by Vinen (1957a,b,c,1958) and later investigated numerically by Schwarz (1988). In both cases,the dissipation of flow energy is through the motion of vortex lines; its rateper unit mass can be assumed to be (Stalp et al., 1999, 2002; Vinen, 2000)

E = −ν(κL)2, (38)

where κ2L2 is an effective total mean square vorticity and the parameter νis believed according to various models to be approximately constant fora given temperature and type of flow. This formula is the quantum analogof the classical expression Equation (37). As we will show below, the effi-ciency of the vortices in dissipating energy in the T = 0 limit, expressedthrough the ‘effective kinematic viscosity’ ν, is different for these tworegimes of turbulence.


To measure the values of ν, one can monitor the free decay of homoge-neous tangles of both types. In any tangle, the quantum energy associatedwith the quantised flow on length scales r < � is Eq = γL/ρs (per unitmass), where the energy of vortex line per unit length is γ = Bρsκ

2 andB ≈ ln(�/a0)/4π is approximately constant. This is the same energy thatis shown in the quantum range k > 1/� (including both noncascadingvortex skeleton and Kelvin-waves) in Figure 27. If the total energy ismainly determined by Eq, from Equation (38), we arrive at the late-timefree decay

L = Bν−1t−1. (39)

This type of turbulence, without any motion on classical scales, willbe called ultraquantum (in the literature one can find, e.g., ‘Vinen tur-bulence’ (Volovik, 2003, 2004) and ‘random’ or ‘unstructured’ vortextangle). For 4He, κ4 = 2π�/m4, a0 ∼ 0.1 nm and B ≈ 1.2, while for 3He-B,κ3 = 2π�/(2m3), a0 ∼ 13−65 nm and B ≈ 0.7.

Now suppose that the tangle is not random but structured due tothe presence of flow on classical length scales r > �, and the additionalenergy of this classical flow Ec is much greater than Eq. This type of tur-bulence will be called quasiclassical (e.g., ‘Kolmogorov turbulence’ and‘structured’ or ‘polarised’ vortex tangle). For the Kolmogorov spectrumbetween wavenumbers k1 and k2 (k1 � k2), while the size of the energy-containing eddy stays equal to the size of container h (that is, k1 ≈ 2π/h),the late-time free decay follows (Stalp et al., 1999)

L = (3C)3/2κ−1k−11 ν−1/2t−3/2. (40)

Eventually, when the energy flux from the decay of classical energyEc will become smaller than that from the quantum energy Eq (whichshould typically happen when only a couple of vortex lines are left inthe container, that is, L ∼ h−2), the quasiclassical regime will cross overto the ultraquantum one, so L ∝ t−3/2 decay will be replaced by L ∝ t−1.Deviations from the described scenario are also possible if some frac-tion of the classical energy is stored in non-cascading ‘thermal spectrum’(L’vov et al., 2007a).

In this Section, we first review the relevant experimental techniquesof detecting and generating turbulence. Next we describe the more recentdevelopments at the very lowest temperatures. The results of these mea-surements will be discussed in detail in the last part of this section, wherewe conclude with a general discussion of the significance of the observedtemperature dependence of ν.


4.1.2 Techniques of Measuring the Vortex Line Density L

The attenuation of second sound in superfluid 4He, which is proportionalto the total length of vortex lines, has been the prefered technique ofmeasuring L in the temperature range 1–2 K (Barenghi and Skrbek, 2007;Niemela et al., 2005; Stalp et al., 1999; Vinen, 1957a) since Hall and Vinen(1956); however, it cannot be extended below 1 K. Very recently, scatteringof negative ions off vortex lines has been succesfully utilised at tempera-tures 80 mK–1.6 K (Walmsley et al., 2007a), showing good agreement withprevious measurements of ν at overlapping temperatures (Stalp et al.,2002). Actually, the first use of ions for the observations of a vortex tanglewas reported almost 50 years ago (Careri et al., 1960), and numerous fur-ther experiments revealed their potential for investigating turbulence insuperfluid 4He (see the reviews by Tough (1982) and Donnelly (1991) formore references on early studies).

It is worth mentioning possible alternative techniques of detectingturbulence in 4He in the T = 0 limit that are currently being developed,such as calorimetric (Ihas et al., 2008) and others (Vinen, 2006). The designof towed grid for turbulence generation, suitable for experiments at T <

0.5 K in 4He, is presently being attempted by McClintock at Lancaster andIhas at Gainsville. Optical vizualisation of individual vortex lines in the tur-bulent tangle (Bewley et al., 2006, 2008; Rellergert et al., 2008; Sergeev et al.,2006; Zhang et al., 2005) (see the contribution by Van Sciver and Barenghi,2008, in this volume) is another promising development, although theprospects of extending this technique down to at least T = 0.5 K seem dis-tant. As we mentioned in Sections 2 and 3, computer simulations havecontributed a great deal to our understanding of certain processes in thedynamics of tangles and of the Kelvin-wave cascade at T = 0 in particular(Hänninen et al., 2007a; Kivotides et al., 2001; Schwarz, 1988; Tsubotaet al., 2000; Vinen et al., 2003) (see also the contribution by Tsubota andKobayashi, 2008 in this volume); however the proper modelling of theinertial cascade pumped at large (quasiclassical) length scales and dissi-pated at short (quantum) scales requires a considerable range of lengthscales and hence remains a challenge.

4.1.3 General Properties of Injected Ions in Liquid Helium

Injected ions (see reviews (Borghesani, 2007; Donnelly, 1991; Fetter, 1976;Schwarz, 1975)) are convenient tools to study elementary excitations andquantised vortices in superfluid helium. They were used to detect vortexarrays in rotating helium (Careri et al., 1962; Yarmchuk et al., 1979), quan-tised vortex rings (Rayfield and Reif, 1964) and vortex tangles (Careri et al.,1960) in 4He. To create a negative ion, an excess electron is injected intoliquid helium, where it self-localises in a spherical cavity (‘electron bub-ble’) of radius ∼2 nm at zero pressure. To create a positive ion, an electron


is removed from one atom, which results in a positively-charged clus-ter ion (‘snowball’) of radius ∼0.7 nm. These objects are attracted to thecore of a quantised vortex, resulting in trapping with negligible escaperate provided the temperature is low enough (T < 1.7 K for negatives andT < 0.4 K for positives in 4He at zero pressure). Because of this interac-tion, both species have been used a great deal to study quantised vorticesalthough negative ions are often more convenient, especially if one wantsto cover a wider range of temperatures. The binding energy of a nega-tive ion trapped by a vortex is about 50 K (Donnelly, 1991). In superfluid3He, thanks to much larger diameters of vortex cores, this binding ismuch weaker; hence there were no observations of ion trapping on vortexcores yet. Still, the anisotropy of ions motion in 3He-A made it possi-ble to use negative ions to detect textural changes around vortex cores(Simola et al., 1986).

In this section, we focus on negative ions in superfluid 4He. Theirmobility is limited by scattering thermal excitations and is hence rapidlyincreasing with cooling. Below some 0.8 K, it is so high that they quicklyreach the critical velocity for creation of (depending on pressure andconcentration of 3He impurities) either rotons (and then move with theterminal Landau velocity ∼60 m/s, shedding off rotons continuously) orquantised vortex rings (and then get trapped by such a ring and movewith it as a new entity, a charged vortex ring) (Berloff and Roberts, 2000a,b;Hendry et al., 1988, 1990; Winiecki and Adams, 2000). Above 1 K, the trap-ping diameter σ of an ion by a vortex line is inversely proportional to theelectric field, σ ∝ E−1. In a field E = 33 V/cm, it is decreasing with temper-ature from σ = 100 nm at T = 1.6 K to σ = 4 nm at T = 0.8 K (Ostermeierand Glaberson, 1974).

Rayfield and Reif (1964) produced singly-charged, singly-quantisedvortex rings in 4He at T = 0.4 K and, by measuring the dependence of theirself-induced velocity v ∝ κ3K−1, Equations (41) and (42), on their energyK, confirmed the value of the circulation quantum κ = 1.00 × 10−3 cm2s−1.Schwarz and Donnelly (1966) investigated the trapping of charged vortexrings by rectilinear vortices in a rotating cryostat and found the trappingdiameter to be of order of the ring radius, meaning that the interaction isbasically geometrical. They wrote: “quantised vortex rings are very sen-sitive “vortex-line detectors,” making them suitable probes for a numberof problems in quantum hydrodynamics”. Guenin and Hess (1978) usedcharged vortex rings to detect turbulent vortex tangles in 4He at T = 0.4 K,created by forcing a jet of superfluid through an orifice at supercriticalvelocities.

While charged vortex rings are more convenient than free ions becauseof their much greater trapping diameter, their dynamics is more peculiar(Figure 29; all figures in Sections 4.1–4.3 represent the Manchester exper-iments (Walmsley et al., 2007a; Walmsley and Golov, 2008a)). The energy


1.0

0.8

Tim

e of

flig

ht (

s)0.6

0.4

0.2

00 5

Electric field (V/cm)

10

HorizontalVertical

1.5

1.0

Cur

rent

(pA

)

0.5

00 1 Time (s) 2

E 5 20 V/cm Horizontal Vertical

3

15 20 25

FIGURE 29 Time of flight (corresponding to the leading edge of the pulse ofcollector current shown in inset) of charged vortex rings as a function of electric fieldat T = 0.15 K in both the horizontal and vertical direction. The solid line shows thetimes of flight for rings with initial (as injected into the drift volume) radii andenergies of 0.53mm and 21 eV, calculated using Equations (41), (42) and (43).

K, velocity v and impulse P of a quantised vortex ring of radius R with ahollow core of radius a0 and no potential energy in the core are (Donnelly,1991; Glaberson and Donnelly, 1986)

K = 12ρsκ

2R(

ln8Ra0

− 2)

, (41)

v = κ

4πR(

ln8Ra0

− 1)

and (42)

P = πρsκR2. (43)

By integrating these equations, one can calculate the trajectories of thecharged vortex rings subject to a particular electric field (Walmsley et al.,2007b).

4.1.4 Measuring the Vortex Line Density L by Ion Scattering

In the absence of vortex lines, the ions would propagate either freely(these dominate at T > 0.8 K) or riding on small vortex rings (dominateat T < 0.6 K). Without other vortex lines in their way, the time of flight of


1.0

0.8

0.6

0.4

0.2

15 s

Cur

rent

to r

ight

col

lect

or (

pA)

12.5 s

200 sT 5 1.60 KT 5 0.90 K

T 5 0.15 K 0.15 K, 0.31 s, 20 V/cm0.90 K, 0.10 s, 10 V/cm1.60 K, 0.50 s, 20 V/cm

00 0.5 1.0

Time after start of injection (s)1.5 2.0 2.5 3.0

FIGURE 30 Examples of current transients produced by short pulses for threedifferent temperatures (the temperatures, pulse durations and mean driving fields areindicated in the legend). The solid lines show the transients without a vortex tangle inthe ions’ path, while the dashed ones represent the transients suppressed by thevortex tangle which has been decaying a specified time (indicated near curves) afterstopping generation. The charge carriers are either free ions (T = 1.60 K andT = 0.90 K) or charged vortex rings (T = 0.15 K). The electronics time constant is 0.15 s;hence the time of arrival of the fastest peak (T = 0.90 K) cannot be resolved.

both species over a particular distance is a well-defined function of tem-perature, electric field and initial radius of the attached vortex ring (if any).Hence, after injecting a short pulse of such ions, a sharp pulse of currentarrives at the collector as shown in Figure 30. The interaction with otherexisting vortices, that happened to be in the way of the propagating ions, ischaracterised by a ‘trapping diameter’ σ (Ostermeier and Glaberson, 1974;Schwarz and Donnelly, 1966) and leads to the depletion of the pulses ofthe collector current. This is used to measure the average density of vortexlines L between the injector and collector through

L(t) = (σh)−1 ln(I(∞)/I(t)). (44)

To calibrate the value of σ for either free negative ions or ions trapped ona vortex ring at different temperature and electric field, one can measurethe attenuation of the pulses of the collector current when the cryostat isat continuous rotation at angular velocity � (thus having an equilibriumdensity of rectilinear vortex lines L = 2�/κ) as in Figure 31. This is bestdone in the direction perpendicular to the rotation axis.

At low temperatures in quantum cascade Kelvin-waves of a broadrange of wavelengths are excited; however the main contribution to thetotal length L converges quickly at scales just below � (Kozik and Svistunov,


1.0� � 0.13 μm

� � 0.72 μm

� � 1.56 μm

1.60 K, 20 V/cm0.78 K, 100 V/cm0.50 K, 20 V/cm

0.8

0.6

0.4I/I (

Ω�

0)

0.2

00 0.5 1.0

Ω (rad/s)

1.5 2.0

FIGURE 31 Dependence of pulse amplitude on the angular velocity of rotation, �,in examples of calibration measurements. The temperatures, driving fields andtrapping diameters σ are indicated. The charge carriers are either free ions(T = 1.60 K) or charged vortex rings (T = 0.78 K and T = 0.50 K).

2008b). Hence, a probe ion with the trapping diameter σ � �, movingat a speed (v ∼ 10 cm/s) much greater than the characteristic velocitiesof vortex segments (∼κ/� < 10−1 cm/s), should sample the full length L.Experimental measurements with different σ = 0.4–1.7 mm (using chargedvortex rings in a range of driving electric fields) indeed produce consistentvalues of L.

There are evidences that small concentrations of trapped space charge(when the ratio of trapped space charge density n to the vortex line lengthL does not exceed n/L ∼ 105 cm−1) do not affect the tangle dynamics(Walmsley and Golov, 2008b). For example, experiments generating qua-siclassical tangles at T ≥ 0.7 K by a jet of ions (and hence resulting in asubstantial trapped space charge) revealed that the late-time decay is iden-tical to that for the tangles generated by an impulsive spin-down to restwithout injecting any ions. Still, one might worry that the very presenceof trapped charge, as it interacts with the beam of probe ions, might affectthe result of measuring L. Hence, in order not to introduce extra turbu-lence and not to contaminate the tangle with any ions, the measurementwas performed by probing each realisation of turbulence only once, aftera particular waiting time t during its free decay. Then the contaminatedtangle was discarded and a fresh tangle generated.

4.1.5 Design of Ion Experiment

The cell used in the Manchester experiments had cubic geometry withsides of length h = 4.5 cm. A schematic drawing of the cell is shown in


Left tip

Bottom tipBottom electrode

with grid

Topgrid

Top collector

Rightcollector

Ω

Right electrodewith grid

Left electrodewith grid

Top electrode

FIGURE 32 A side cross-section of the Manchester cell. The distance betweenopposite electrode plates is 4.5 cm. An example of the driving electric field is shownby dashed lines. Such a configuration was used in all the Manchester experimentsdescribed in this section. It was calculated for the following potentials relative to theright electrode: left electrode at −90 V, side, top and bottom electrodes at −45 V, andthe right electrode and collector at 0 producing a 20 V/cm average driving field in thehorizontal direction. To inject ions, the left tip was usually kept at between −500 to−350 V relative to the left grid. When ions traveling vertically across the cell (injectedfrom bottom tip and detected at the top collector) were required, the potentials onthe electrodes were rearranged as appropriate.

Figure 32. The relatively large size of the cell was important to enhance theefficiency of ion trapping and the time resolution of vortex dynamics, andwas really instrumental to ensure that the continuum limit � � h holdseven when the vortex line density drops to as low as just L ∼ 10 cm−2.This also helped ensure that the presence of the walls, which might accel-erate the decay of turbulence within some distance ∼�, does not affect thedynamics of the turbulent tangle in the bulk of the cell. In order to probethe vortex densities along the axial and transverse directions, there weretwo independent pairs of injectors and collectors of electrons. Both injec-tors and collectors were protected by electrostatic grids, enabling injectingand detecting pulses of electrons. The injectors were field emission tipsmade of 0.1 mm diameter tungsten wire (Golov and Ishimoto, 1998). Thethreshold for ion emission was initially � −100 V and −210 V for thebottom and left injectors, respectively; however, after over two years ofalmost daily operation, they changed to some −270 V and −520 V. Thefact that the two injectors had very different threshold voltages helped


investigate the dependence (or rather lack of it) of the radius of initialcharged vortex rings on the injector voltage. The six side plates (electrodes)that make up the cube can be labelled ‘top’, ‘bottom’, ‘left’, ‘right’, andtwo ‘side’ electrodes. The top, bottom, left and right electrodes had cir-cular grids in their centres. All grids were made of square tungsten meshwith period 0.5 mm and wire diameter 0.020 mm, giving a geometricaltransparency of 92%. The grids in the bottom, left and right plates had adiameter 10 mm and were electrically connected to those plates. The gridin the top plate had diameter of 13 mm and was isolated from the top plate.The injector tips were positioned about 1.5–2 mm behind their grids. Thetwo collector plates were placed 2.5 mm behind their grids and were typ-ically biased at +10 to +25 V relative to the grids. Further details can befound in Walmsley et al. (2007b).

4.1.6 Tangles Generated by an Impulsive Spin-down

This novel technique of generating quasiclassical turbulence, suitable forany temperatures down to at least 80 mK, relied on rapidly bringinga rotating cubic-shaped container of superfluid 4He to rest (Walmsleyet al., 2007a). The range of angular velocities of initial rotation � was0.05–1.5 rad/s. In classical liquids at high Re, spin-down to rest is alwaysunstable, especially at high deceleration and in axially asymmetric geome-tries. It is known that, within a few radians of initial rotation upon animpulsive spin-down to rest, a nearly homogeneous turbulence developswith the energy-containing eddies of the size of the container. In a cylindri-cal container, centrifugal and Taylor-Görtler instabilities usually break upat the perimeter, thus facilitating slowing down of the outer region of theinitially rotating liquid. Simultaneously, because of the Ekman pumpingof nonrotating liquid into the central axis via the top and bottom walls,the central core of the initial giant vortex slows down too (Donnelly, 1991).In a cubic container, the turbulence becomes homogeneous much faster.Some residual rotation of the central region in the original direction mightstill survive for a while but, as we show below, the generated turbulenceis in general pretty homogeneous.

This behaviour is well-documented for classical liquids (van Heijst,1989), and one expects similar processes to occur in a superfluid liquidproviding the process of initial multiplication of vortices does not affectthe dynamics. As spin-down experiments always begin from already exist-ing dense rectilinear vortex arrays of equilibrium density L = 2�/κ, andrapid randomisation and multiplication of these vortices is expected dueto the lack of axial symmetry of the container, as well as surface pinning(friction) in the boundary region, these seem to be sufficient for the super-fluid to mimic the large classical turbulent eddies. In Figure 33, we showthe four different stages of the evolution from a vortex array to a decaying


(1) (2) (3) (4)

FIGURE 33 Cartoon of the vortex configurations produced by spin-down in theexperimental cell (side view) at different stages. (1) Regular array of vortex lines duringrotation at constant � before deceleration. (2) Immediately after stopping rotation(0 < �t < 10), turbulence appears at the outer edges but not on the axis of rotation.(3) After about 30rad of initial rotation (�t ∼ 30), 3D homogeneous turbulence iseverywhere. (4) Then (�t ∼ 103) the 3D turbulence decays with time. Shaded areasindicate the paths of probe ions when sampling the vortex density in the transverse(Lt, 3) and axial (La, 4) directions.

homogeneous tangle upon an impulsive spin-down to rest, which are inagreement with the observations outlined below.

Before making each measurement, the cryostat was kept at steady rota-tion at the required � for at least 300 s before decelerating to full stop, thenwaiting a time interval t and taking the data point. Then the probed tan-gle was discarded and a new one generated. Hence, different data pointsrepresent different realisations of the turbulence. The deceleration was lin-ear in time taking 2.5 s for � = 1.5 rad/s and 0.1 s for � = 0.05 rad/s. Theorigin t = 0 was chosen at the start of deceleration.

4.1.7 Current-Generated Tangles

An alternative technique of generating turbulence, by a jet of injected ionsthat does not require any moving parts in the cryostat, has been devel-oped (Walmsley and Golov, 2008a) (Figure 34). In early experiments withinjected ions at low temperatures (Bruschi et al., 1966, Bowley et al., 1982) itwas observed that a pulse of negative ions through superfluid 4He leavesbehind a tangle of vortices. Walmsley et al. (2008a) found that the proper-ties of these tangles can be quite different. As we explain below, the tanglesgenerated after long injection at high temperatures possess the propertiesof developed quasiclassical turbulence while those produced after shortinjection at low temperatures have signatures of random tangles with littlelarge-scale flow.

At high temperatures T > 0.7 K, while moving relative to the normalcomponent, the ions experience viscous drag through scattering of excita-tions – this means that the ions entrain the normal component along. Theycan also get trapped by an existing vortex line (provided T < 1.8 K), afterwhich the vortex line will be pulled along with the current of such ions.


(1) (2) (3) (4)

FIGURE 34 Cartoon of the vortex configurations produced by a pulse of injectedions at T < 0.5 K in the experimental cell (side view) at different stages. (1) < 1 s: apulse of charged vortex rings is injected from the left injector. While most make it tothe collector as a sharp pulse, some got entangled near the injector. (2) ∼5 s: thetangle spreads into the middle of the cell. (3) ∼20 s: the tangle has occupied allvolume; from now on, it is nearly homogeneous (as probed in two directions). (4) Upto 1000 s: the homogeneous tangle is decaying further. The shaded areas indicate thetrajectories of ions used to probe the tangle along two orthogonal directions.

The existing vortices and hence the superfluid component, through theaction of mutual friction, will be pulled by the already entrained normalcomponent too, and vice versa. Then one can expect the injected currentto produce a large-scale jet-like flow of liquid helium, which is an efficientway of driving quasiclassical turbulence at large scales. This is similar tojet flow through an orifice – one of the traditional means of generatingturbulence in classical liquids (Chanal et al., 2000). On the other hand,at low temperatures the ions always bring small vortex rings along, andhence, upon formating a tangle, can directly pump up the vortex length Lwithout introducing substantial large-scale flow – at least for not too longinjections.

4.2 Experimental Results

4.2.1 Quasiclassical Turbulence Generated by a Spin-Down to Rest

At 1 K < T < 2 K, the L ∝ t−3/2 free decay was monitored by second soundfor towed grid turbulence in Oregon (Stalp et al., 1999). The recent spin-down experiments at Manchester are in good agreement with them atoverlapping temperatures. At T < 1 K, only scattering of ions off vortexlines has been used to measure L so far.

In Figure 35, the measured densities of vortex lines along the horizontalaxis (transverse, Lt) are shown for four different initial angular velocities�. During the transient, which lasts some ∼100�−1, Lt(t) goes through themaximum after which it decays eventually reaching the universal late-time form of L ∝ t−3/2. For � ≥ 0.5 rad/s, the values of L at maximumwere too high to be detected. The initial vortex densities at steady rotation,L = 2�/κ, are shown by horizontal lines. Similarly, the densities of vortexlines measured along the vertical axis (axial, La) are shown in Figure 36.


104

0.5 rad/s Ω 5 1.5 rad/s

0.15 rad/s

Lt,

cm2

2 t 23/2

0.05 rad/s

T 5 0.15 K

103

102

101

100 101 102

t (s)103 104

FIGURE 35 Lt(t) at T = 0.15 K for four values of �. The average driving fields used for� = 1.5 rad/s: 5 V/cm (◦), 10 V/cm (�), 20 V/cm (�), 25 V/cm (�). At other values of �,the electric fields used were either 20 V/cm (0.05 rad/s) or 10 V/cm (0.5 and0.15 rad/s). The dashed line shows the dependence t−3/2. The horizontal bars indicatethe initial vortex densities at steady rotation, L = 2�/κ, at � = 1.5 rad/s, 0.5 rad/s,0.15 rad/s and 0.05 rad/s (from top to bottom).

104

Ω 5 0.5 rad/s

0.15 rad/s 1.5 rad/s

La,

cm

22

t 23/2

0.05 rad/s

T 5 0.15 K

103

102

101

100 101 102

t (s)103 104

FIGURE 36 La(t) at T = 0.15 K for four values of �. The average driving field used was20 V/cm in all cases except at 0.5 rad/s where both 10 V/cm (�) and 20 V/cm (�) wereused. The dashed line shows the dependence t−3/2. The horizontal bars indicate theinitial vortex densities at steady rotation, L = 2�/κ, at � = 1.5 rad/s, 0.5 rad/s,0.15 rad/s and 0.05 rad/s (from top to bottom).


2.0

1.5

Ωt 5 1.5

Ωt 5 0

Ωt 5 6

Ωt 512

During rotationTime after stopping:

t 5 1 st 5 4 st 5 8 sLong time limit

Cur

rent

to to

p co

llect

or (

pA)

1.0

0.5

0

0 1 2Time (s)

3 4 5

FIGURE 37 Records of the current to the top collector, injected from the bottominjector as a 0.1 s-long pulse at the time t after an impulsive spin-down from� = 1.5 rad/s to rest. T = 0.15 K, E = 20 V/cm.

To illustrate what is happening near the vertical axis of the containerat different stages of the transient after a spin-down, in Figure 37 we showfive records of the current to the top collector arriving after a short (0.1 s)pulse of probed ions was fired from the bottom injector. Each is character-istic of a particular configuration of vortex lines near the rotational verticalaxis of the container during the transformation from an array of parallellines to a homogeneous decaying vortex tangle. One can see three differentcharacteristic times (vertical dashed lines in Figure 37) of arrival of ionsvia different means. The first peak at ≈ 0.4 s (determined by the time con-stant of the current preamplifier, 0.15 s) corresponds to the ions trapped onthe rectilinear vortex lines which can slide along those lines very quickly,provided those lines are continuous from the injector to collector as dur-ing steady rotation. The second peak at ≈ 1 s corresponds to the coherentarrival of the ballistic charged vortex rings from the bottom to the top. Thethird broad peak at times ∼3 s (but with a long tail detectable until ∼40 s)corresponds to the charge trapped on the vortex tangle and drifting withthe tangle. Hence, we have the following regimes (and curves in Figure 37)labelled by the time t after the spin-down:

1. �t ≤ 0, steady rotation. The nearly equal first and second peaks tell thatthere exist a vortex array without disorder (otherwise the second peakwould get suppressed and the third would appear).


2. �t = 1.5. The first peak gets enhanced three-fold while the second oneis still there (and no third peak) – meaning more trapped ions can nowreach the collector along the rectilinear vortex lines while there is notmuch turbulence in this region yet.

3. �t = 6. The first peak has disappeared in favour of the second onewhich has got broadened – at this stage, the rectilinear vortices shouldhave become scrambled in the Ekman layers near the top and bot-tom walls/grids while the ballistic charged vortex rings are still thedominant transport of charge.

4. �t = 12. Now both fast peaks have disappeared completely while thethird broad peak has emerged – this means that a turbulent tangle hasfinally reached the axial region.

5. �t → ∞. The sharp second peak has recovered but all others van-ished – after the turbulence has decayed only ballistic charged vortexrings carry the charge, neither the rectilinear array nor turbulent tanglecontributes to the transport any more.

In Figure 38, the measured densities of vortex lines along the hori-zontal, Lt, and vertical, La, axes are shown by solid and open symbols,respectively. To stress the scaling of the characteristic times with the ini-tial turnover time �−1 and the universal late-time decay ∝ t−3/2, the datafor different � are rescaled accordingly. We can see that at all � the

105

104

103

102

101

Ωt

T 5 0.15 K

LΩ2

3/2 ,

cm

22

s3/2

1021 100

0.05 rad/s

0.15 rad/s

0.5 rad/s

1.5 rad/s

101 102 103 104

t 23/2

FIGURE 38 �−3/2Lt(t) (filled symbols) and �−3/2La(t) (open symbols) versus �t forfour values of � at T = 0.15 K. The straight line ∝ t−3/2 guides the eye through thelate-time decay.


transients are basically universal. Immediately after deceleration, Lt jumpsto ≈ 104�3/2 cm−2s3/2, indicating the appearance of the TBL at the perime-ter, while La is stable at Li ≈ 2�/κ. Only at �t ≈ 3, the latter starts to grow,signalling the destruction of the rotating core with vertical rectilinear vor-tices. After passing through a maximum at �t = 8 and �t = 15, Lt andLa merge at �t ∼ 30 and then become indistinguishable. This implies thatfrom now on the tangle density is distributed nearly homogeneously. Thescaling of the transient times with the turnover time �−1 indicates thattransient flows are similar at different initial velocities�, which is expectedfor flow instabilities in classical inviscid liquids. Eventually, after�t ∼ 100,the decay takes its late-time form L ∝ t−3/2 expected for quasiclassicalisotropic turbulence, whose energy is mainly concentrated in the largesteddies bound by the container size h but homogeneous on smaller lengthscales. We hence assume that the turbulence in 4He at this stage is nearlyhomogeneous and isotropic, and can apply Equation (40) to extract theeffective kinematic viscosity ν. The crossover to the L ∝ t−1 regime at latetime has never been observed, probably because the measured values ofL never dropped below 10 cm−2 � h−2 ∼ 0.05 cm−2.

At 0.08 < T < 0.5 K, the measured L(t) were independent of tempera-ture. In Figure 39, we compare the transients Lt(t) at low (T = 0.15 K) andhigh (T = 1.6 K) temperatures. The prefactor in the late-time dependenceL ∝ t−3/2 at T = 0.15 K is almost an order of magnitude larger than that forT = 1.6 K. This implies that at low temperatures, the steady state inertialcascade with a saturated energy-containing length and constant energyflux down the range of length scales requires a much greater total vor-tex line density. This means that in the T = 0 limit the effective kinematicviscosity ν is approximately 70 times smaller than at T = 1.6 K (providedC ≈ 1.5 and k1 ≈ 2π/h are independent of temperature).

At all temperatures, the transients Lt(t) after a spin-down are, in firstapproximation, universal, that is, the timing of the maximum is the same,�t ≈ 8, and the amplitudes of the maximum are comparable. This sup-ports our approach to turbulent superfluid helium at large length scalesas to an inviscid classical liquid, agitated at large scale and carrying an iner-tial cascade down the length scales, independent of temperature. On theother hand, as the temperature increases and interaction with the viscousnormal component becomes stronger, changes in the shape of transientsmight be expected. Indeed, one can see that the slope of L(t) after themaximum but before reaching the ultimate late-time decay L ∝ t−3/2 (thatis, for 10 < �t < 100) is changing gradually with increasing temperaturefrom being less steep than t−3/2 at T = 0.15 K to more steep than t−3/2 atT = 1.6 K (Figure 39). At temperatures around T = 0.85 K (Figure 41), itnearly matches t−3/2, thus making an erroneous determination of ν pos-sible by taking this part of the transient for the late-time decay L ∝ t−3/2.Indeed, in the first publication (Walmsley et al., 2007a), parts of some


105

104

103

102

101

T 5 0.15 K

T 5 1.6 K

L tΩ

23/

2 , c

m2

2s3/

2

Ωt1021 100 101 102 103 104

0.05 rad/s

0.15 rad/s

0.5 rad/s

1.5 rad/s

t 23/2

FIGURE 39 �−3/2Lt (t) versus �t for T = 0.15 K (filled symbols) and T = 1.6 K (opensymbols). Dashed and solid lines ∝ t−3/2 guide the eye through the late-time decay atT = 1.6 K and 0.15 K, respectively.

transient at T = 0.8−1.0 K for as early as �t > 15 were occasionally usedto be fitted by L ∝ t−3/2 that often resulted in overestimation of the valuefor ν. To rectify this, we have refitted the data sets used in the originalpublication as well as subsequent measurements with L ∝ t−3/2 for spin-downs using the following rules: for T ≤ 0.5 K, only points for �t > 300were used, between 0.5 K and 1.0 K only points for �t > 150 were usedand at T > 1.0 K, only �t > 75 were used. This resulted in slight system-atic reduction of the extracted values of ν(T) at temperatures 0.8–1.2 Kfrom those published in Walmsley et al. (2007a); what looked as a rathersteep drop in ν(T) at 0.7–0.8 K, now occurs at 0.85–0.90 K and is somewhatreduced in magnitude.

4.2.2 Quasiclassical Turbulence Generated by an Ion Jet

As mentioned above, ion jets can produce quasiclassical tangles decay-ing as L ∝ t−3/2 (Walmsley and Golov, 2008a). Examples of the decay ofsuch tangles alongside with those obtained by a spin-down at T = 1.6 Kand T = 0.8 K are shown in Figures 40 and 41, demonstrating goodquantitative agreement between the late-time decays. At temperatures0.7–1.6 K, the late-time decays of quasiclassical turbulences generated bythese two different techniques were identical within the experimentalerrors.


105

104

103

102

1 10 100

T 5 1.60 KIon jetSpin-down from 1.5 rad/sSpin-down from 0.5 rad/s

L(c

m2

2 )

t (s)

t 23/2

FIGURE 40 Free decay of a tangle produced by a jet of free ions from the bottominjector (•) (Walmsley and Golov, 2008a), as well as by an impulsive spin-down to rest(�) (Walmsley et al., 2007a) from 1.5 rad/s and 0.5 rad/s, at T = 1.60 K. All tangleswere probed by pulses of free ions in the horizontal direction. The line L ∝ t−3/2

corresponds to Equation (40) with ν = 0.2 κ.

T 5 0.85 KIon jetSpin-down from Ω 51.5 rad/s

1 10 100t (s)

t 23/2

105

104

103

102

L(c

m2

2 )

FIGURE 41 Free decay of a tangle produced by a jet of free ions from the bottominjector (•) (Walmsley and Golov, 2008a), as well as by an impulsive spin-down to rest(Walmsley et al., 2007a) from 1.5 rad/s, at T = 0.85 K. All tangles were probed bypulses of free ions in the horizontal direction. The ion jet data are the average of ninemeasurements at each particular time, but the spin-down data show individualmeasurements.


4.2.3 Ultraquantum Turbulence Generated by a Jet ofCharged Vortex Rings

In the temperature range 0 < T < 0.5 K, all tangles produced by a pulseof injected current of duration 0.1–1 s, intensity 10−12–10−10 A and in thedriving field 0 − 20 V/cm revealed the late-time decay L ∝ t−1, all withthe same prefactor (Figure 42) (Walmsley and Golov, 2008a). This univer-sality of the prefactor for all initial vortex densities is a strong argument infavour of the random character of the tangles whose decay is described byEquation (39) (ultraquantum turbulence). During the injection, the tangleoriginates near the injector, presumably as the result of colliding manycharged vortex rings all of the same radius R ≈ 0.5 mm, and then spreadsinto the centre of the cell in 3–5 s, eventually filling all container and becom-ing nearly homogeneous after ∼20 s. This can be seen at the transient inFigure 42 (∗) where the tangle was initiated at the bottom injector butthen sampled along the horizontal axis of the container. The dynamicsof the tangle spreading was found to be independent of the driving field0–20 V/cm.

The tangles produced by the bottom and left injectors had very nearlyidentical late-time decays L ∝ t−1 (Figure 42). The corresponding values ofν/κ, inferred using Equation (39), are 0.120 ± 0.013 and 0.083 ± 0.004. Asthe scattering diameters σ of charged vortex rings produced only by the

100 101101

102

102

103

103

t (s)

L(c

m2

2 )

Inject: bottom (0.3 s, 10 V/cm)Inject: bottom (0.3 s, 20 V/cm)Inject: left (0.1 s, 20 V/cm)Inject: bottom (0.3 s, 20 V/cm), probe: left

t21

FIGURE 42 Free decay of a tangle produced by beams of charged vortex rings ofdifferent durations and densities, T = 0.15 K. The injection direction and duration anddriving field are indicated. Probing with pulses of charged vortex rings, of duration0.1–0.3 s were done in the same direction as the initial injection, except in one case(∗). The line L ∝ t−1 corresponds to Equation (39) with ν = 0.1κ.


left injectors could be calibrated directly in situ on the arrays of rectilinearvortex lines at steady rotation, the absolute value of ν = 0.08κ for thesetangles probed along the horizontal axis is treated as more reliable.

4.3 Discussion: Dissipation in Different Types of Turbulence

4.3.1 Dissipation in 4He at Different Temperatures

Let us now summarise what is known about the dissipation of varioustypes of turbulence in superfluid 4He at different temperatures. As therelevant parameter, we plot in Figure 43 the effective kinematic viscosityν (as defined in Equation (38)) which is a function of the mutual fric-tion parameter α(T) but can be different for different types of flow. Onecan see that at high temperatures, T > 1 K, all experimental points grouparound ∼0.1 κ, give or take a factor of two. The fact that different types ofexperiments seem to produce slightly different absolute values andtemperature dependences might have various reasons. First, experimentaltechniques rely on the means of calibrating the sensitivity to the absolutevalue of L and on the knowledge of other parameters in the model. Thecalibration is normally performed on an array of rectilinear vortex linesin the direction perpendicular to the lines but not on tangles of vortexlines. For example, recent refined calculations (Chagovets et al., 2007) sug-gested to correct all previous second sound measurements of L on isotropictangles by a factor of 3π/8 ≈ 1.2, which we apply here. Second, the exactvalues and dependence on temperature of the effective parameters relat-ing L(t) to ν (such as B ≈ 1.2 in Equation (39) and C ≈ 1.5 and k1/h ≈ 2πin Equation (40)) are not known, thus complicating precise comparisonof ν for different temperatures and types of flow. Third, at T > 1 K, thenormal component can also become turbulent (Barenghi et al., 2002; L’vovet al., 2006b; Schwarz and Rozen, 1991; Vinen, 2000), hence its velocityfield might vary for different means of generating turbulence.

Theoretically, one can consider two limiting cases: either completelylocked (vn = vs) turbulent flow of both components or an absolutely lam-inar normal component (vn = 0 in a local reference frame) that slowsdown the segments of quantised vortices moving past it. The former isfavourable at high mutual friction α(T) and low viscosity of the normalcomponent ηn(T) (that is, at the high-temperature end) and is expectedto be described by the kinematic viscosity ηn/ρ (dotted line in Figure 43),where ρ = ρn + ρs, while the latter is favourable at low α and high ηn(that is, at the low-temperature end) and is expected to follow the pre-dictions of mutual-friction-controlled models which assume a laminarnormal component (e.g., line-connected stars and asterisks for randomtangles in Figure 43). The experimental situation is obviously somewherein between (and the measured ν should probably tend to the smaller value


0 0.5 1.0

�n /� �n /�n

T (K)

�(T ):

/

1.5

Ultraquantum:CVRs

counterflowultrasound

simulationsQuasiclassical:

ion jetspin-down

towed girdfit KS2008b

2.01023

1022

100

1021

1025 10241023 1022 1021

FIGURE 43 The effective kinematic viscosity ν(T ) for various types of turbulence in4He. (i) Quasiclassical turbulence: ν(T ) inferred from the L ∝ t−3/2 dependent freedecay of tangles using Equation (40) and produced by impulsive spin-down (opentriangles Walmsley et al. (2007a)); an ion jet (open circles Walmsley and Golov,2008a); towed grids of two different designs (open squares Niemela et al., 2005; Stalpet al., 2002). The solid curve is theory (Eq. 45) by Vinen and Niemela, 2002, and thedashed curve is a fit to the Walmsley et al. (2007a) data (open triangles and circles) byKozik and Svistunov (2008b). (ii) Ultraquantum turbulence: ν(T ) inferred usingEquation (39) from L ∝ t−1 dependent free decay of tangles produced by collidingcharged vortex rings (filled up and down triangles, Walmsley and Golov, 2008a);counterflow (filled right-pointing triangles, Vinen, 1957b and filled squares, Schwarzand Rozen, 1991); ultrasound (filled diamonds, Milliken et al., 1982); computersimulations (line-connected stars, Tsubota et al., 2000); as well as from the analysis ofthe measured tangle density in applied counterflow (filled left-pointing triangles,Vinen, 1957b); and computer simulations (line-connected asterisks, Schwarz, 1988).Note that in simulations ( and ∗), the normal component was artificially clamped tolaminar flow, while in most experiments at T > 1 K it is involved in turbulent motionsto different extent. The values of the kinematic viscosities, ηn/ρn of the normalcomponent and ηn/(ρn + ρs) of the ‘coupled normal and superfluid components’ areshown by dotted lines (Donnelly and Barenghi, 1998).

of those predicted by the two limits), although the particular means ofdriving might tip the balance towards either of the limits. This might beone reason why different techniques show systematic disagreement in thetemperature range above 1 K.

One can speculate that quasiclassical turbulent flow generated bymechanical stirring is less prone to such uncertainties since both the super-fluid and normal large-scale flows are generated simultaneously and are


probably locked. On the other hand, the generation of quasi-randomtangles was attempted by a variety of techniques: those continuouslypumped by counterflow in narrow channels might have the normal com-ponent more laminar, while those with freely decaying turbulence in widechannels were shown to evolve towards coupled superfluid and normalturbulence at high temperatures (Barenghi and Skrbek, 2007), especiallyso if created with ultrasound which equally pumps turbulence in bothcomponents. In any case, on approaching T = 1 K from above, when thenormal component progressively becomes more viscous and less dense,all experimental and numerical values of ν(T) for quasi-random tanglesseem to converge, which supports these reasonings.

At temperatures below 1 K, the values of ν(T) appear to split rapidly:those for ultraquantum turbulence apparently stay at nearly the samelevel, ∼0.1 κ, as at high temperatures, while those for quasiclassical tur-bulence keep decreasing until the zero-temperature limit of ∼0.003 κ. Themean free path of excitations in 4He rapidly increases with decreasingtemperature and becomes comparable with � ∼ 100 mm at 0.7 K and withthe container size h = 4.5 cm at 0.5 K. At these temperatures, the normalcomponent is too viscous to follow the turbulent superfluid compo-nent and hence the convenient model of a laminar normal componentbecomes adequate here (in this respect, this regime of 4He is similar tothat in superfluid 3He-B below 0.4 Tc, discussed in previous sections). It isindeed comforting that numerical simulations of tangles initiated at shortscales (without large-scale turbulence (Tsubota et al., 2000)) are in goodagreement with experiments for random tangles at these temperatures(Walmsley and Golov, 2008a). As yet, there are no satisfactory computersimulations of homogeneous quasiclassical turbulence in the limit of zerotemperature.

Between 1.1 K and 1.6 K, the values of ν(T) for quasiclassical turbulencegenerated with towed grids of different designs (Niemela et al., 2005; Stalpet al., 1999), impulsive spin-down to rest and an ion jet are reasonablyconsistent between each other and show an increase from ν = 0.05 κ to0.2 κ. There was an early attempt (Stalp et al., 1999) to relate these valueswith the kinematic viscosity ηn/ρ = 10−4 cm2/s ≈ 0.1 κ, assuming lockedturbulence of the superfluid and normal components. However, it wassoon realised (Stalp et al., 2002) that the contribution of quantised vortexlines to dissipation controlled by mutual friction and reconnections is alsoimportant. This certainly dominates below 1.2 K, where ηn/ρ begins toincrease with cooling (hence, making the normal component effectivelydecoupled from vortices on short length scales) while the measured ν(T)

decreases with cooling.In the mutual-friction-dominated regime and provided the normal

component is only locked to the superfluid velocity at scales > �, theactual dissipation is through the self-induced motion of vortex lines


relative to the normal component, whatever the kinetics of the quantumcascade. In this regime, a segment of a vortex line bent at radius R andhence moving at self-induced velocity v ∼ BκR−1 dissipates energy atthe rate ∼ρsαB2κ3R−2 per unit length, where B ≈ 1.2 is introduced inEquation (39). This approach leads to the formula (Vinen, 2000; Vinen andNiemela, 2002)

ν = sB2c22ακ . (45)

The multiplier s is meant to account for the degree of correlation betweenthe motion of individual segments of vortex lines and the surround-ing superfluid in a structured quasiclassical tangle (s ≈ 0.6), comparedto a random tangle of the same density L (s = 1). In the simulations ofcounterflow-maintained tangles in the local induction approximation bySchwarz (1988), the measure of the relative abundance of small-scale kinkson vortex lines, c2

2 = L−1 〈R−2〉, was found to increase from 2 to 12 as α

decreased from 0.3 to 10−2 with decreasing temperature from 2 to 1 K. Thisformula nicely agrees with the simulations by Tsubota et al. (2000). It alsocaptures the trend and magnitude of ν(T) for all experimental quasiclas-sical tangles as well as for the random tangle maintained by counterflow(Vinen, 1957a) (where the normal component is expected to be nearly lam-inar); although to fit it to the data of quasiclassical tangles in Figure 43(solid line), we used s = 0.2, apparently owing to partial locking betweenthe flows in the normal and superfluid components.

As vortex segments with smaller radii of curvature R lose their energyfaster, for a given energy flux down the length scales, there exists a dis-sipative scale λq below which the cascade essentially cuts off (Kozik andSvistunov, 2008b; L’vov et al., 2006b; Svistunov, 1995; Vinen, 2000). Fora developed nonlinear Kelvin-wave cascade (λq � �), with the ampli-tude spectrum bk ∝ k−6/5 (Kozik and Svistunov, 2004), this implies that〈R−2〉 ∼ �−2/5 λ

−8/5q . In this sense, the Schwarz’s parameter c2

2 ∼ (�/λq)8/5

quantifies the range of wavelengths involved in the quantum cascadeand becomes another quantum analog of the classical Reynolds number:Re = (h/λcl)

4/3. With decreasing α, the value of λq progressively decreasesuntil, at T < 0.5 K (Kozik and Svistunov, 2008b; L’vov et al., 2006b; Vinen,2001), it reaches the wavelength λph ∼ 3 nm at which phonons can be effec-tively emitted. Below this temperature, as the dissipation of Kelvin-wavesdue to mutual friction becomes negligible compared to the emission ofphonons, ν(T) is independent of temperature. However, in the presenceof the quantum cascade at larger wavelengths, it is the cascade’s kineticsthat controls the energy flux, and it might happen that the vortex lengthL already saturates at higher values of α, thus signalling the onset of theregime with ν(T) = const as T → 0, Equation (38).


Let us now discuss the specific models for the low-temperaturebehaviour of ν(T). For the dissipative mechanisms that involve individ-ual quantised vortices, such as those maintained by vortex reconnections,simple considerations (Vinen and Niemela, 2002) usually predict ν to beof order κ. However, more involved models, such as that of a ‘bottleneck’for quasiclassical turbulence, might suggest smaller values (L’vov et al.,2007a). If certain configurations of vortices are inefficient in transferringenergy down to short wavelengths (e.g., either there exists a noncascad-ing part of the spectrum or reconnections are less efficient in polarisedtangles), the resulting accumulation of an extra contribution to L leads toa reduction of the parameter ν. This is what is observed for quasiclassicaltangles in 4He on approaching the T = 0 limit.

An important question remains: how is the energy of classical eddiespassed over to the shorter quantum length scales? Two main mechanismsare currently discussed: (i) the excitation of Kelvin-waves through purelynonlinear interactions in classical eddies or (ii) vortex reconnections (eachleaving a sharp kink on both vortex lines, hence effectively redistribut-ing the energy to smaller length scales). Both predict an increase in L(T)

at constant energy flux down the cascade (i.e., the decrease of ν(T)) withdecreasing temperature. The former considers the accumulation of extranoncascading quasiclassical vorticity at length scales above � due to thedifficulty in transferring energy through wave numbers around �−1 ifreconnections do not ease the process (referred to as the bottleneck modelin Section 3) (L’vov et al., 2007a). The latter mainly associates the excessvortex length L with the contributions from the new self-similar struc-tures produced by vortex reconnections on length scales shorter than �

when vortex motion becomes progressively less damped at low temper-atures (Kozik and Svistunov, 2008b). This model relies on four types ofprocesses: reconnections of vortex bundles, reconnections of neighbour-ing vortices, self-reconnections of a vortex, and nonlinear interactions ofKelvin-waves, which bridge the energy cascade from the Kolmogorov tothe Kelvin-wave regimes. It can be successfully fitted to the experimentalν(T), as seen in Figure 43. In short, both models predict an enhancementof vortex densities but on different sides of the crossover scale �.

Finally, we may ask why is ν(T =0) larger for ultraquantum (random)tangles than for quasiclassical tangles in the T = 0 limit? Whatever themodel, this means that, for the same total density of vortex lines L, the rateof energy dissipation in random tangles is larger. Within the bottleneckmodel (L’vov et al., 2007a), this seems straightforward as these tangles haveno energy on classical scales at all. Within the reconnections/fractalisationscenario (Kozik and Svistunov, 2008b), this is explained by the fact thatreconnections in partially polarised tangles are less frequent and lessefficient. In the framework of this latter model, to explain why there isapparently no temperature dependence of ν(T) for ultraquantum tangles


below T = 1 K (although no experimental data exist for the range 0.5–1.3 K), one should assume that for this type of tangle all significant increasein the total line length L caused by the fractalisation with decreasingtemperature has already occurred above 1 K.

4.3.2 Comparing Turbulent Dynamics in 3He-B and 4He

It would be instructive to provide a comparison of turbulent dissipation insuperfluid 4He and 3He-B. Unfortunately at present, with only few mea-surements on varying types of flow, this remains a task for the future. TheLancaster measurements on the decay of an inhomogeneous tangle gen-erated with a vibrating grid by Bradley et al. (2006) revealed a L ∝ t−3/2

late-time decay (Fisher and Pickett, 2008). Assuming that the origin of thet−3/2 dependence is the same as in quasiclassical turbulence with a satu-rated energy-containing length, and that this length is equal to the spreadof the turbulent tangle over a distance of 1.5 mm, using Eq. (40) they extractthe temperature-independent value of ν ≈ 0.2 κ at 0.156 < T/Tc < 0.2.However, it is questionable whether the condition for the applicability ofEq. (40), that the classical energy Ec dominates over the quantum energyEq, is met for this turbulence. From their recent measurements of the initialvelocity of mean large-scale flow, u ∼ 0.5 mm/s (Bradley et al., 2008), andvortex density, L ∼ 104 cm−2, we estimate Ec ∼ u2/2 ∼ 10−3 cm2/s whileEq ∼ 0.7κ2L ∼ 10−2 cm2/s; i.e. Ec � Eq even at the early stage of decay.Hence, it is possible that the dynamics of the decay of their localizedtangles is that of an ultraquantum turbulence but accelerated due to thediffusion or emission of vortex rings into space; thus the decay L(t) issteaper than ∝ t−1. More experiments with grid turbulence in 3He-B aredesirable, which could shed light on the temperature dependence of tur-bulent dynamics, for example, if there is a similar decrease in ν(T) belowα ∼ 10−3 as measured for 4He or to provide an independent characteri-sation of the energy-containing length scale. It would also be importantto reproduce the results for 4He in the T = 0 limit with another methodof generating turbulence (for example, by means of towed or vibratinggrids).

The Helsinki experiments on the propagating vortex front (Section 3)provide a measure of dissipation via the front velocity Vf. The observedrelative enhancement of the rate of decay below 0.4 Tc (α < 1) proves theefficiency of the turbulent cascade in dissipating the energy of large-scaleflow. The leveling off of this rate below 0.28 Tc (α < 10−2) is qualitativelysimilar to that observed in 4He at T < 0.5 K (α < 10−5), and might hintat the T = 0 limit in the inertial cascade. However, the respective valuesof α, at which the leveling-off occurs, differ substantially. Various reasonscan be given to explain this difference, for example that different typesof flow have their own specific ν(α) dependence (evidence for this is the


difference in the ν(T) values of quasiclassical and random tangles). Forinstance, to test the interpretation of the temperature dependence of Vf(T)

on approaching the T → 0 limit, given in Section 3.5.6 in terms of the prox-imity of the outer and quantum crossover scales, it would be instructiveto conduct similar measurements in containers of different radii.

In front propagation, the energy dissipation rate is proportional toVf(T), while in experiments described in this section the rate is pro-portional to ν(T). Thus, it is interesting to compare the temperaturedependences of Vf(T) and ν(T) in the two superfluids. We plot in Figure 44the effective kinematic viscosity ν/κ and the normalised velocity of thepropagating front vf = Vf/�R as a function of mutual friction dissipationα. Note that the role of turbulence comes into play in a very differentmanner in these two types of flow: as nearly homogeneous and isotropicturbulence in 4He in the Manchester experiments and, roughly speak-ing, via the deviation from laminar vortex front propagation, owing toturbulent excitations in 3He-B in the Helsinki experiments.

It is striking that for α < 0.1 (i.e., the regime where developed turbu-lence becomes possible) and before leveling off at T → 0, the data for allexperiments and numerical simulations discussed here approach a simi-lar slope of ∼α0.5. For ν(α) in the spirit of Equation (45), this implies thatthe parameter c2

2 is roughly proportional to α−0.5 – even well below thevalues of α of the calculations by Schwarz (1988). On a finer scale, particu-lar models of matching the classical and quantum cascades, as discussedin the previous subsection, might produce specific dependences of c2(α),which perhaps can be tested in future experiments in more detail, providedthat the quality of data is improved.

We do not attempt to compare the absolute values of ν/κ and vf inFigure 44. Instead, we concentrate on the α-dependences, especially onthe values of α at which the dissipation rates level off as T → 0. The factthat the measured 3He-B data level off at higher values of α might sug-gest that the quantum cascade in 3He-B is not as developed as in 4He andthat the ultimate dissipative mechanism, independent of α in the α → 0limit, is stronger and takes over at larger length scales in 3He-B. Indeed,the numerically calculated velocity of front propagation vf(α), which doesnot explicitly incorporate any excess dissipation beyond mutual frictiondissipation and hence is equally suitable for 3He-B and 4He, is steadilydecreasing with decreasing α, following the same trend as the experimen-tal and numerical ν(T) for 4He. In contrast, the measured vf(α) levels offbelow α ∼ 1 · 10−2, which seems to imply that the T = 0 regime prevails atα < 1 · 10−2 in 3He-B, while this happens only below α < 1 · 10−5 in 4He.The fermionic nature, the large vortex core diameter, and the much lowerabsolute temperatures lead to different inherent dissipation mechanismsin 3He-B. For example, the energy loss during each reconnection event canbe substantial in 3He-B but not in 4He. This is supported by the fact that the


1021

1021

1022

1022

10231024102510261027

�100

100

/

,Vf/

ΩR

FIGURE 44 Normalised measures of dissipation rate, the effective kinematicviscosity ν and the normalised velocity of the propagating vortex front vf = Vf/�R,as a function of mutual friction dissipation α in 4He and 3He-B: (i) ν(T ) forquasiclassical turbulence in 4He: spin-down, open triangles (Walmsley et al., 2007a);ion jet, open circles (Walmsley and Golov, 2008a); towed grid, open diamonds(Niemela et al., 2005; Stalp et al., 2002); (ii) ν(T ) for ultraquantum turbulence in 4He:colliding charged vortex rings, filled triangles (Walmsley and Golov, 2008a); numericalsimulations, line-connected stars (Tsubota et al., 2000), and asterisks (Schwarz, 1988).(iii) ν(T ) estimates for ultraquantum turbulence in 3He-B: dashed line (Bradley et al.,2006). (iv) vf in 3He-B: experiment, open squares (Eltsov et al., 2007); and numericalsimulations, line-connected squares (Section 3).

values of νmeasured in Lancaster for 3He-B with a0 ∼ 30 nm (at P = 12 bar)level off (i.e., branch away from the common trend of ν(α) marked by thenumerically calculated dependences) at larger values of α ∼ 3 · 10−2 thanthe measured front velocity vf for a0 ∼ 16 nm (at P = 29 bar) atα ∼ 1 · 10−2.We conclude by noting that further work on turbulence in different typesof flow in both 4He and 3He-B is highly desired.

4.4 Summary: Decay of Turbulence on Quasiclassical andUltraquantum Scales

The newly developed technique of measuring the density L of a vor-tex tangle by the scattering of charged vortex rings of convenient radius∼1 mm has made it possible to monitor the evolution of tangles in super-fluid 4He down to below 0.5 K, that is, deep into the zero-temperaturelimit. The dynamics of two very different types of tangles can be studied.


Quasiclassical tangles, mimicking the flow of classical liquids on largelength scales, were generated by an impulsive spin-down from angularvelocity � to rest of a rotating cubical container with helium. After a tran-sient of duration ∼100/�, the turbulence becomes nearly isotropic andhomogeneous and decays as L ∝ t−3/2, as expected for turbulence posess-ing a Kolmogorov cascade of energy from the energy-containing eddies ofconstant size, set by the container size h. This was studied in isotopicallypure 4He in a broad range of temperatures of 80 mK–1.6 K, correspondingto the range of mutual friction α from ∼10−10 to 10−1. Identical resultswere obtained for the free decay of quasiclassical tangles generated by acentral jet of ions although only at temperatures above 0.7 K so far.

Alternatively, nonstructured (ultraquantum) tangles of quantised vor-tices, that have little flow at scales above the inter-vortex distance andhence no classical analog, can be obtained by colliding many small quan-tised vortex rings of radius R � � at temperatures below 0.5 K, that is inthe zero-temperature limit. These tangles took about 10 s to spread froman injector into all experimental volume of size h = 4.5 cm, followed byfree decay with the universal dynamics L ∝ t−1, independent of the initialconditions. The relatively fast rate of spreading is indeed surprising andmight be due to the small polarisation (mean velocity) of the tangle.

Quantitative measurements of the free decay for both types of tanglesallow the extraction of the ‘effective kinematic viscosities’ ν – the flow-specific parameter linking the rate of energy dissipation with the totaldensity of vortices L. It turned out that at high temperatures T > 1 K, wherethe quantum cascade is not well developed, the values of ν for both types ofturbulence are comparable. However, in the zero-temperature limit, wherethe dissipation can only take place at very short length scales � �, to whichthe energy can only be delivered by a cascade of nonlinear Kelvin-waveson individual vortex lines, the saturated value of ν for quasiclassical tur-bulence seems to be smaller than that for ultraquantum turbulence. Onthe microscopic level, this is most probably related to the fact that in thepresence of quasiclassical eddies the partial mutual alignment of vortexlines in ‘bundles’ slows down the process of exciting Kelvin-waves for agiven vortex density L. For instance, this mutual alignment is expectedto reduce the frequency and efficiency of vortex reconnections, which arebelieved to be the defining process of the dynamics of nonstructured (ultra-quantum) tangles as they fuel the Kelvin-wave cascade. At present, twomicroscopic models are developed to describe the energy transfer fromclassical to quantum scales (cascades) in quasiclassical tangles: one rely-ing on reconnections and the other assuming that additional noncascadingvortex density helps maintain the continuity of the energy flux down tothe dissipative length scale.

We have also compared the rates of dissipation as a function of mutualfriction in superfluid 4He and 3He-B. Both similarities and differences


in the behaviour are spotted and discussed. However, a final quantitativeverdict on the role of specific mechanisms cannot be delivered at this stage,pending further development of theoretical models and of experiments ondifferent types of turbulent flow in both superfluids.

5. SUMMARY

In the last five years, we have witnessed important advances in theunderstanding of the appearance, growth and decay of different typesof superfluid turbulence, especially in the fundamentally important limitof zero temperature where the intrinsic processes within the superfluidcomponent set the dynamics of the tangle of quantised vortices. As theseare absolutely undamped on a broad range of length scales down towavelengths of order 10 nm–1 mm, depending on the type of superfluid,a principally new microscopic dynamics emerges – set by the instabilitiesof individual vortices, their reconnections and 1-dimensional cascades ofenergy from nonlinear Kelvin-waves (wave turbulence) along individ-ual vortex lines. Amazingly, even in this limit, various turbulent flowsbehave classically on large length scales, and often the observed rate ofdecay is no different from that at high temperatures – owing to the factthat the energy-containing and dissipative lengths and times are wellseparated.

However, the discrete nature of quantised superfluid vorticity alsobecomes important in such processes as the growth of an initial seed vor-tex tangle out of a single-vortex instability, the efficiency of dissipatingthe energy of large-scale eddies through short-wavelength Kelvin-waveson individual vortex lines, or the dynamics of nonstructured tanglespossessing no large-scale flow. In the helium superfluids these underly-ing processes are separately observable and are ultimately expected tobecome the corner stones of a detailed theoretical framework. This shouldmake it possible to develop a consistent picture of turbulence in heliumsuperfluids, which describes the nonlinear turbulent dynamics of dis-crete line vortices in a macroscopically coherent quantum liquid of zeroviscosity.

ACKNOWLEDGEMENTS

This work is supported by the Academy of Finland (grants 213496,124616 and 114887), by ULTI research visits (EU Transnational Access Pro-gramme FP6, contract RITA-CT-2003-505313) and by EPSRC (UK) (grantsGR/R94855 and EP/E001009).


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